The most widespread piece of fanon in Harry Potter is that Hogwarts uniforms include skirt/trousers, shirts and house-specific ties, with robes on top (which are also sometimes lined with their house colors). This comes from the movies - the canon uniform consists of plain black robes, with no house markings.
This should be obvious to anyone who's read Chamber of Secrets, since Harry and Ron wouldn't have mistaken a Ravenclaw for a Slytherin had half her uniform been blue.
Also in book canon, the robes are complete outfits unto themselves and not traditionally worn over Muggle shirts and pants, as a middle-aged wizard's ignorance of Muggle clothing in Goblet of Fire makes clear.
For some reason, said skirt on the fanon Hogwarts uniform for girls is almost always depicted as a short mini-skirt in fanart (in spite of it being around knee-length in the movies, most likely due to the Rule of Sexy).
There was a widespread notion that Ginny's name was short for Virginia (a notion that originated from The Draco Trilogy), until JKR revealed that her full first name was actually the much rarer Ginevra. It was not unheard of for long-running fics to go back and change it, or else quietly retcon it and hope the readers didn't notice. (In one story, Harry complains to Ginny, "You're the person who let me think her name was Virginia for four years before finally revealing it was really Ginevra...")
People continue to believe that "Percy" is short for "Percival," but Order of the Phoenix uses "Percy" for the official record of Harry's hearing.
The spell called 'tempus' that tells the time is pure fanon, despite what many fanfic authors think.
As are 'wards'. The word is used exactly once in the books, and that one does not refer to protective magic. In the books spells of this kind are called "protective enchantments/charms/spells."
Nowhere are Moony, Wormtail, Padfoot, and Prongs actually referred to as the Marauders. In fact, with regards to the map they made, it's the singular: "Marauder's", not "Marauders'".
Word of God claims that they referred to themselves as such, but that was after it was widespread fanon. By then it was the easiest way out.
A similar one is the fandom insistence that Fred and George routinely refer to themselves/each other as Gred and Forge. This is a one-off joke in the first book, when their mother made them Christmas sweaters with their initials on them. Apparently people really liked it.
Annoyingly, most fanfic gets it backwards. They have sweaters with their correct initials; the joke is that George doesn't know the rest of their names. He also apparently thinks Percy's name is 'Prefect'.
They also usually finish each other's sentences in fanfics, which happens less than a handful of times in Canon. This misconception was probably popularized by a trailer for the Prisoner of Azkaban movie featuring them switching off every few words when explaining the Marauder's Map.
In canon Molly had two brothers named Fabian and Gideon, who died fighting Death Eaters. A common Fanon belief is that Fred and George's initials were meant to honor them, and/or that "Fabian" and "Gideon" are their middle names. Another purely Fanon belief is that Fabian and Gideon were twins.
If you do the math, it seems that Charlie and Tonks were in the same Hogwarts year. From that developed the idea that they were very close—possibly even childhood friends—who eventually dated before deciding they wereBetter as Friends. Often Tonks has some Affectionate Nickname for him. You'd think this would've been brought up in the gang's conversation about Tonks joining the family in Phoenix. Interestingly, this usually isn't used to break up the ship between Remus and Tonks, but more for background for those two characters.
Similarly, Percy and Oliver Wood are often paired together or imagined as friends, apparently just because they were in the same year and same House.
Rowling: I think he was a user and a narcissist and I think someone like that would use it, would use the infatuation. I don't think that he would reciprocate in that way, although he would be as dazzled by Dumbledore as Dumbledore was by him, because he would see in Dumbledore, 'My God, I never knew there was someone as brilliant as me, as talented as me, as powerful as me. Together, we are unstoppable!' So I think he would take anything from Dumbledore to have him on his side.
It's generally taken for granted that Snape is Draco's godfather. It's stated nowhere in canon, but most people believe it to be as true as the fact that Sirius is Harry's godfather. The 6th book however points out that Draco and Snape aren't all that close, Draco regards Snape as an interloper who is trying to take Lucius' place in Voldemort's organization and his life, and the 7th Book has Snape confess to Dumbledore that Draco no longer looks up to him after Lucius was imprisoned.
Also, the idea that the Malfoys and Snape knew each other was around long before the sixth book, which was the first time they even interacted with each other. Before JKR made it canon, the only connection between Snape and Lucius was that they were both Death Eaters and Snape's tendency to praise and shield Draco.
Oh, that crazy Head Boy and Girls' dormitory which only exists in fanfiction! This has been around so long that the specifics of the room have been pretty much codified. It's inside a tower. It normally contains a common room with two separate dorm rooms, one for the Head Boy and one for the Head Girl. There's always only one bathroom to ensure hilarious sexual hijinks. It's often entered through a painting with the subject of the painting and the password being something vaguely romantic. You'd think Dumbledore would have something better to do than to be The Matchmaker!
Note that canon disproves this: in Prisoner of Azkaban Percy is Head Boy (and his girlfriend is even Head Girl!), but he still sleeps in Gryffindor Tower.
"Magical Cores", which are basically power levels, show up in about ninety percent of HP fanfiction and only there; there is not the slightest hint in canon at anything like that.
If anything, people thinking that every wizard or witch has some sort of innate genetic magical potential are missing the point of the entire series.
This also tends to be linked to the fanon that cores or magical abilities can be "suppressed" in some way, usually to justify Harry and/or Hermione becoming even more powerful and having noncanon abilities. Sometimes this is also the case with Neville, to explain why he's so incompetent (because of course it can't just be that he was a kid who wasn't great at magic and needed a little extra training to get better). Almost always, the suppression is revealed to be done by Dumbledore, for some unexplained, vaguely sinister reason. Usually, it's to make whoever's magic is being suppressed reliant on Dumbledore's protection, guidance etc., and therefore a more compliant pawn.
Occasionally, the fanon says that Neville's magic is (unintentionally) suppressed by a massive Obliviation charm placed on him to make him forget the torture of his parents. It's so powerful that it makes his memory bad... so he's unable to do magic well as he can never remember the magic because the Obliviation spell keeps erasing it in case any of it inadvertently triggers memories of his parents torturing.
There's also "Ancient and Noble Houses", and their magical Lord rings, which Harry, as "Lord Potter" (and sometimes Neville) will inherit. The only time a character talks about them, it's Sirius Black referring to his family, and he was mocking their pureblood ideals.
It's not uncommon for those of high rank to be members of the Wizengamot, and for the good guys to engage in power games with Lucius Malfoy. In canon, it's not clear how members are selected, nor does Lucius (as far as we know) even have a seat on the Wizengamot-in fact, we know that he wasn't a member as of Harry's fifth year.
Additionally, the idea that "Lords", "Ladies" and "Heirs" (mind the capitals, everyone) have special privileges at Hogwarts, including Lords' Quarters. In fact, Hogwarts is a fantastical version of the very real British public school system, one of the points of which was to educate the paying public, noble and commoner alike, on an equal footing. To this end, whatever privileges you had, whether you were the sitting Duke of Norfolk or the Prince of Wales, ended once you were a student enrolled at this school. Your only privileges were whatever you earned through school, such as, in the case of Hogwarts, being prefect, Head Boy/Girl, or quidditch captain. Indeed, traditionally, such schools had an instituted system of "fagging", the purpose of which was to "knock the nonsense out of" students, especially those of genteel or noble birth, and to remind them that, while they attended school, they were no better than any other students.
On another related note, the idea that each school founder had an heir, and that these heirs, singly or in concert, could assert authority over the school, even revoking the school charter if they so desired. Except no school has ever been run along those lines, particularly not the British public schools on which Hogwarts is, in part, modeled.
A number of Snapefics have him referring to or thinking of his Slytherin students as his "little snakes", prompting a Heh Heh, You Said "X" response from many readers.
Since Hermione's parents' names are never mentioned, fanfic writers have instead invented their own. Roger and Helen crop up frequently; Dan and Emma were common for a while, but have since dropped in popularity, probably after everyone figured out how cringe-inducingly meta it was. Then there was a wave of Rose and Hugo (since Hermione's just as creative as Harry is) and even a bit of Wendell and Monica (Hermione's probably clever enough not to give her parents their real names as false names).
Some of the very minor student character have been used as O.C. stand-ins enough to develop fanon personalities. Daphne Greengrass is an Ice Queen, but free from pure-blood prejudice and at least slightly more sympathetic than Pansy. Tracey Davis is Daphne's best friend and is more easygoing. Padma Patil is the opposite of Parvati and possibly the second brightest student of her year after Hermione. Blaise Zabini is a suave player type. Theodore Nott is a Jerkass and when the other Slytherins are portrayed more positively, he'll be the voice of pure-blood prejudice.
Since the only things known about Blaise before Half-Blood Prince were his name and that he is in Slytherin (from the sorting ceremony in Philosopher's Stone), many fansnote except in French-speaking countries, where Blaise is a male-only name, unlike gender-neutral names like Camille or Dominique assumed he was a she — and no one ever guessed he was black. Most Fem!Blaises were porcelain-skinned beauties of obvious Italian heritage.
Theodore Nott is also usually portrayed in fanfic as a close friend of Malfoy, despite Word of God saying that he's a loner who doesn't feel the need to fit in with Malfoy and his cronies.
This has at least some canon backing, since Nott is an old friend of Malfoy's, and one of the only people he considers an equal. Rowling even wrote a scene with the two of them in Malfoy Manor, but couldn't fit it anywhere. He's just not one of Malfoy's goons.
Theodore, when given character development, ends up either the cruelest Death Eater out of all the Slytherins, or a sympathetic character that secretly sabotages the Death Eaters from the inside.
Although we see enough of Pansy Parkinson to know that she's the Alpha Bitch, we know nothing about her life outside of Hogwarts. The usual fanon is that she's from a lesser pure-blood family known for sucking up to people like the Malfoys. Thus, Pansy is a social climber hoping to become a Malfoy by marriage, which fits the usual view of her character.
Views on Astoria Greengrass differ. Since she's a very minor character who's never mentioned in the books and is only known for being Daphne's younger sister and Draco's future wife, she's ripe for tons of interpretations. A popular one is that after the war, she played the gentle girl to Draco's brooding boy and helped him get back on his feet.
Then there's the idea that when the officiator said Bill and Fleur were "bonded for life" he meant it literally, and wizard weddings involve magic that forces the couple never to cheat on each other or divorce.
Voldemort's line, "There is no good and evil. There is only power, and those too weak to seek it," gets quoted a lot, as though it were the official creed of the Death Eaters or something—which might be possible, but is never mentioned to be the case. The line is, after all, only spoken once in the entire series.
Tom Riddle's orphanage was run by nuns. This fanon was completely justifiable, even likely, until Mrs. Cole was introduced in Half-Blood Prince. Nevertheless, it still persists in the world of fanfiction.
To show that Harry knows more than he should (especially in Peggy Sue fics), the Sorting Hat is sometimes referred to as "Adrian", even though a name for it is never mentioned in the books.
The notion that Professor McGonagall went to school with Tom Riddle and was an Auror during the War against Grindelwald crops up frequently. This is probably because of her actress, Maggie Smith, being far older than McGonagall was intended to be, despite there being only a year between the two, since the films were made a decade after the books' time frame. According to Pottermore, McGonagall was born nine years after Tom Riddle in 1935, and therefore only started Hogwarts in 1947, two years after Grindelwald's defeat.
Also popular is the theory that she was a champion quidditch player (usually a beater or chaser), that she was a notorious prankster (or a notorious Ice Queen bookworm, or sometimes, if the author is feeling clever, a notorious Ice Queen who was secretly the forerunner to the Weasley twins), and that she was best friends with Pomfrey and Sprout. Pottermore confirms that she was indeed a chaser and that she is, indeed, a friend of Sprout.
Another near-universal belief is that Susan Bones is a redhead with (after puberty, at least) an impossibly voluptuous figure. Her hair colour is briefly mentioned in the Sorting in the first book, but her figure is never brought up.
A common theme in "X Reads the Harry Potter books" (and possibly in other types of fics, as well) is that Remus has an obsession with (or at greatly enjoys) chocolate. This may stem from the fact that he was seen carrying an enormous chocolate bar (as a remedy for dementor encounters) in his first appearance, even though he was never said to have a particular fondness for it. It may have caught on thanks to The Shoebox Project, a fic that inspired quite a lot of common Marauder tropes; in it, chocolate is randomly Remus' Trademark Favorite Food—an idea that many fanfic writers copied.
Masteries, apprenticeships and other levels and certifications of magical education beyond Hogwarts are entirely a fan creation; JK Rowling has explicitly stated that there's nothing more to learn after seventh year.
Speaking of masteries, "schoolmaster" is an archaic term for a (male) teacher and Snape's title of "Potions Master" is clearly supposed to reference that, but this hasn't stopped fandom from coming up with all kinds of other interpretations.
The idea that the snake Harry set free from the zoo in the first book later became Nagini. This was a popular fan theory before the end of the series. After the series ended, it was sort of forgotten that the theory had never actually been confirmed and it slipped into the realm of fanon. There's even an associated quote attributed to Rowling, but God Never Said That. In fact, it doesn't make any sense anyway, since the snake at the zoo is a boa constrictor and Nagini is a venomous snake. Also, the snake at the zoo had a male voice in the film and Nagini is a female snake.
Relatedly, there's the popular theory that the snake was actually a captive animagus. This, along with the above theory, seems to have its roots in the common observation that the snake speaks Spanish ("Thanksss, amigo.") instead of Portuguese, despite supposedly being Brazilian. (The fact that "amigo" means "friend" in both languages seems to be quietly ignored.) note Brazil's primary language is Portuguese, not Spanish; and Rowling is definitely fluent in Portuguese, as she lived and worked in Portugal for several years before writing Harry Potter. Both theories are a common way to explain the discrepancy, coming to the conclusion that the snake was of human-like intelligence, and simply trying to pass itself off as a common boa constrictor from Brazil.
The Draco Trilogy either created or codified the notion that the Malfoy family home is called "Malfoy Manor". Pretty soon it became fanon that every rich pure-blood family lives in a mansion called "[last name] Manor", even as the canon contradicted this by showing that the Black family, at least, lived at Number 12, Grimmauld Place (don't think this has stopped "Black Manor" from showing up in fanfiction). The name "Malfoy Manor", however, got canonized in Harry Potter and the Deathly Hallows. Whether Cassandra Claire guessed accurately or Rowling just used the fanon name is unclear. (It's worth noting, however, that The Draco Trilogy uses the terms "Malfoy Manor" and "Malfoy Mansion" interchangeably, but only the former name got picked up.)
"Snape Manor" was also a popular fanon before his actual residence was revealed in Half-Blood Prince, despite there being no indication that he was even upper class.
Nearly every depiction of adult!Teddy gives him blue hair, perhaps to match his mom's pink and from the fact that his hair was turquoise in a picture his father showed of him as a baby.
Oh, the fics where all the Marauders live together, even after James is married. One person rightly called it "Lily playing Wendy Bird to a den of Lost Boys". One or more Marauders rooming together for the first year or two out of Hogwarts is plausible, but Lily joining them stretches it beyond the bounds of credibility, especially once she gets married to James.
Some of it might come from the fact that Word of God stated that James financially supported Lupin as well as Lily, due to werewolves having a nearly impossible time finding a job. Still, Lupin living with them is a stretch (even if he's the most sane member of the group).
It's often suggested that wizards use betrothal contracts. This has never been exactly stated, and fanon mainly uses it to justify putting Harry with other females, or to be used to try and force him and Ginny together.
It may have been Grindlewald's phrase, but many assume that Dumbledore, particularly his bashed form, works to obtain his own 'Greater Good'.
Some of it probably stems from the seventh book showing teenage Dumbledore agreeing with Grindlewald in a letter that working for the Greater Good is the best way to go. That aside, acting like he still wants to control the wizarding world ignores the fact that Dumbledore explicitly states that after his sister died, he decided that it was far too dangerous for him to be trusted with power and avoided it (hence his refusal several times over to be Minister of Magic).
On a related note, the idea that, even in canon, Dumbledore has significant political power as Chief Warlock of the Wizengamot, on a par with the Minister for Magic. (In this, some American fans might be thinking of the American system of checks and balances, perceiving Dumbledore as the equivalent of the Chief Justice of the U.S. Supreme Court.) In fact, Word of God says that the Wizengamot is part of the Department of Magical Law Enforcement, which is, in turn, subordinate to the Minister. Additionally, when Dumbledore is on the outs with Fudge during Harry Potter and the Order of the Phoenix, Fudge is easily able to have Dumbledore removed as Chief Warlock. Thus, while Dumbledore is certainly powerful, magically, and has considerable influence, his actual political authority is limited.
If fanon were true, Veela bond with their mates magically, and their reputation as heart breakers came from them being forced by their nature to go after mates with a certain high percentage. If a more compatible (Harry) is known by a Veela (Fleur), she is forced by her nature to leave her old boyfriend (Bill) to do so. This is basically the crux of any pairing with Harry to Fleur or Gabrielle.
There's also the whole issue of male Veela. In canon, we only see Veela women, though it's never explicitly stated that males don't exist. (Presumably they would have to for "part-Veela" to mean anything.) In fanfiction, you get weird ideas like Draco being a male Veela who undergoes his own magical love-bond, usually with either Harry or Hermione.
One known male part-Veela (1/8th to be precise!) does exist in canon; Louis Weasley, Fleur's son with Bill — though whether he has the good looks of the Veela is unknown.
Alternatively, Veelas have a predestined 'mate,' often with the caveat that the Veela will die if they do not have sex by a certain point. Fleur's arc with Bill blatantly proves this false, but that doesn't seem to matter. More often used to force a same-sex pairing into play. Can also be done with werewolves, vampires, animagi, elves, or pretty much any magical demi-human.
The Potters and Longbottoms were never shown to be friends, aside from Harry and Neville, but fanfiction often depicts them as ancient allies and Alice as Harry's godmother and close friend of Lily's. It's quite plausible, particularly since Frank and Alice were also Order members, but there's no canon proof. The only indication of close friendships that the couple could have had is Lily mentioning in a letter to Sirius that she cried all night when she heard that Marlene McKinnon and her family died.
The Weasleys lost their wealth from either an ancient gambling relative, or from fees for a crime, a theme much rarer. Never said to be true.
Magical Guardians are a completely fanon creation, in their expanded role beyond that of a regular Guardian.
Harry having a godmother, who's usually Alice Longbottom.
At times, people in fanfiction depict the dislike of minorities, like Indians and East Asians. This has no merit, seeing as Cho Chang was shown to be very popular and Blaise was 'friends' with Draco.
In fanon, Molly Weasley often has a dislike for Veela. In canon, it was always stated that she disliked Fleur for her personality... until Fleur showed that she wouldn't leave Bill, even when he was scarred by Fenrir Greyback.
Fans have a habit of forgetting basic things, such as the fact that all three Malfoys' one redeeming quality is that they love each other. Why so many fans depict Lucius, and just a bit less often Draco, as heartless wife/mother abusers is a mystery of the world.
Fudge and Umbridge's houses were never confirmed, but they always seem to be Slytherins, though Umbridge is also portrayed as a Hufflepuff fairly often. Given that she's very loyal to Fudge, she fits the bill for Hufflepuff well. Word of God is that she was in Slytherin.
Andromeda is often depicted (particularly in fanart) as a redhead to match her sisters as Blonde, Brunette, Redhead, even though canon states she has brown hair and looks remarkably like Bellatrix.
On the more plausible side, since Bellatrix has the same black hair as her cousins, while Narcissa is blonde and Andromeda is somewhere in between with light brown hair, it's commonly assumed that their father had black hair and their mother was blonde.
Ironically, their father is likely the (former, as he's balding) blond one as Dorea Black, their father's paternal aunt, is blond, meaning it runs in the family.
Harry, Ron and Hermione are almost always collectively known as either "The Golden Trio" or "The Gryffindor Trio." Neither name ever appears in the books (and neither makes much sense anyway, especially considering that they can't be the only three Gryffindors who commonly hang out together).
In next-gen fanfics, Albus Potter, Rose Weasley and Scorpius Malfoy usually form "The Silver Trio," (a term also sometimes used for Ginny, Neville, and Luna) as fanfic authors seem to take it for granted that the three would form a trio much like Harry, Ron and Hermione did, and that Rose and Scorpius would end up dating.
Also in next-gen fanfics, Rose is very often in Ravenclaw and Albus in Slytherin (based on his conversation with Harry at the end of the book.) The second Lily Potter is sometimes a Slytherin as well. A large number of fanworks portray Rose as looking like a red-haired version of Hermione, even though red hair is a recessive gene. Rose is referred to as having her father's hair colour on Pottermore, though (and Hermione could easily carry a recessive redhaired gene on her side, before even getting into the genetic complexities of red hair.)
On a similar note, in fics Hermione pretty much always calls Ron by the name "Ronald" when he bothers her, when in fact in the books she does it very rarely (an instance of it: she calls him Ronald when she's furious at him in Deathly Hallows, when he returns after he left). The only person who does this frequently is his mother. This is more than likely the result of a general trope of mature love interests doing that sort of thing being applied to Harry Potter, despite it never really happening in the books. Also likely influenced by the films, where she does occasionally address him as Ronald.
Voldemort being unable to love due to being conceived under the effects of a love potion. Many take Word of God on the subject and miss the fact that it was merely symbolic.
Rowling: It was a symbolic way of showing that he came from a loveless union – but of course, everything would have changed if Merope had survived and raised him herself and loved him.
What's funny is that if you read the line in context with the question Rowling was answering ("How much does the fact that Voldemort was conceived under a love potion have to do with his non-ability to understand love is it more symbolic?"), you realize that in saying this, Rowling was outright refuting the idea that the love potion caused Voldemort's personality. Rowling Jossed the idea, and got misinterpreted to have confirmed it.
The books depict one Yule Ball, which specifically occurs in conjunction with the Triwizard Tournament. However, High School Dances are good for fanfiction, so why not depict Hogwarts throwing its own every year?
A lot of fanfics describe Dumbledore as the defender of the light.
In the books, there are no politico-cultural factions called "Light" or "Dark", nor is magic ever defined in Force-esque divisions of the same. The most we ever get is a few mentions of "Dark magic."
The abuse the Dursleys heap on Harry is often very exaggerated. Unintentional mistakes are that they regularly starved Harry (according to the first book they didn't, though they did remedy this at the start of the second book) and that there are numerous locks at his door. The latter is truth in the movies, but in the book George and Fred are able to leave Harry's room and fetch his trunk by using a simple hairpin. Also, the bars on Harry's window keep popping up during his later school years even though they got removed during Harry's rescue. There's also the persistent claim that they get money for taking care of Harry which they spend on Dudley.
In the UK, parents and guardians actually do receive money for the children they take care of, so in all likelihood the Dursleys did receive Child Benefits for the presence of Harry in their house. And given established facts about the Dursleys, they probably did spend it all on Dudley.
For that matter, the idea that Vernon is abusive to his entire family. While his treatment of Dudley clearly did the kid no favors, he only ever is affectionate towards his family. In fact, the one thing that nearly makes him throw Harry out of the house for good is learning that Harry has a murderous madman after him, which put Petunia and Dudley's lives in danger. And he only changed his mind because Petunia told him not to throw Harry out.
In addition, many fics treat Harry as being so beaten down that he thinks of himself as a sub-human freak, or else having been regularly beaten by Vernon and Petunia. While Petunia does at one point casually swing a frying pan at Harry, it is never mentioned or even implied that the adult Dursleys ever gave him a full-out beating. Overall, despite his neglect and abuse, Harry is much more well-adjusted and self-confident than many authors would have one believe.
A lot of fanfics claim that Molly and/or Arthur knew the Potters or were somehow friends with them, even though the books say the exact opposite. For starters, all the people who knew Lily and James tend to recognize Harry immediately, because he looks so similar to James (and has Lily's eyes), but Molly and Arthur don't. They are so much older than the Potters that they were certainly not at Hogwarts when they were (Molly explicitly states that the Whomping Willow was planted after she left). It's also explicitly stated that they weren't part of the Order of the Phoenix the first time around (considering that nearly all their children were born during the war they had really good reasons not to be). Molly's brothers were, but they would hardly introduce the members of a secret organization to their family.
Apparently the gold in Harry's vault isn't enough for a lot of fans, since it's often claimed to be just a "trust fund". There's at least another vault with a lot of family heirlooms. And multiple houses, which often come complete with house elves (why the Potters go to Godric's Hollow instead of an ancient manor with nearly unbreakable wards is rarely ever explained, though).
Quite often, it's Dumbledore's doing. The Potter will (or wills if Lily had her own) with the obligatory endless list of wizarding guardians for Harry and the injunction that Harry must never be placed with the Dursleys. This also includes the idea that Dumbledore has obviously suppressed the will (or wills) for his own nefarious reasons.
Likewise fans like to imagine that Harry's wizarding family were essentially the "good guy Malfoys" with large mansions, political clout and illustrious names in wizarding history and that Harry was descended from Godric Gryffindor. The final book and Pottermore revealed that Harry is descended from the famous Peverell family with the Potters themselves being highly mundane and essentially footnotes to wizarding history until Harry came along. The Potter family fortune while sizable is only accumulated from centuries of patents on wizarding products. Most fans never did bother to explain why, if Harry's family was so illustrious, nobody mentions any other Potter other than James?
The Wizengamot is commonly portrayed as a legislative body, usually with hereditary seats, even though in the books, it's never even implied to be anything more than Wizarding Britain's top court.
The Astronomy Tower is often portrayed as the go-to location for after-hours romantic activity, ignoring the fact that it's the one place in the entire school where class is held at night, and ignoring the fact that the entire school is a maze of secret passages and rooms where it would be fairly easy to find a secluded place that isn't on top of a freezing cold tower which is open to the elements.
It's especially bad since we've stumbled across students making out in secret passages in the books, like Ginny and Dean. And Percy and Penny were implicitly making out somewhere in the dungeons in Chamber of Secrets right before Harry and Ron ran across them. And secret potions get made in bathrooms, and Triwiz training takes place in unoccupied classrooms. (Not including the Room of Requirement, but admittedly few people know of that.) The entire series is a string of people hiding inside the castle to do things out of sight, and yet fanon keeps putting couples on the Astronomy Tower.
The idea of custom wands, that somehow if there isn't a single wand that is compatible with a wizard/witch, wandmakers will create a custom wand that suits them perfectly. Usually, if the customer is Harry, then his canonical phoenix holly wand is the result of manipulation by Dumbledore, or to prove he's more powerful than he should be. This obviously ignores rule #1 of wandlore: The wand chooses the wizard. If a wand suited perfectly for a wizard could be made, then they would just make them, not go through hundreds finding a wand that likes them.
This theory probably originated from Fleur mentioning that the Veela hair in her wand came from her grandmother, but this could just be a coincidence on par with Fawkes providing the feathers in Harry and Voldemort's wands.
Traditional (pureblood) wizards celebrate Celtic holidays like Samhain and Beltain and the war is as much a religious war (with purebloods blaming muggleborns for destroying their religion and traditions) as it is about blood-purity. Note, the people who write these stories often are Draco in Leather Pants types.
Sickles and Galleons being made out of pure gold or silver, despite the official conversion rate for pounds and galleons working out such that someone could just keep converting pounds to wizarding coins, melting those coins for their precious metals, and buying more pounds than one started with.
The consequence of breaking an Unbreakable Vow, or the contract of the Triwizard Cup, is almost universally agreed upon in fanon as resulting in a total loss of magic (And yes, this has resulted in at least one fanfic where Voldemort is defeated by being entered into a fake Triwizard Tournament without his knowledge!).
In canon, the consequence for breaking an Unbreakable Vow is explicitly stated to be death, and the one for breaking the Triwizard contract (if there even is one) is never specified.
One very common bit of fanon is that wizards use "Merlin" as a replacement/euphemism for the word "God," leading to expressions like "for Merlin's Sake," "Merlin knows," "honest-to-Merlin" or the big classic, just "Merlin!" While wizards in canon often do mention Merlin, it's always with expressions like "Merlin's beard!" and variations, never just "Merlin." (In fact, wizards in canon do on occasion use "God" as an exclamation or swear; interestingly enough the one to do so most often seems to be Draco Malfoy.)
Some fans have it that everyone in canon, Dumbledore himself included, view Dumbledore as an irrefutable source of wisdom. In actuality, while most people Harry talks to think very highly of Dumbledore, the man himself insists that he is capable of making mistakes and, in fact, that the ones he makes are often the cause of worse consequences than the mistakes of most people. There's also the idea that he deliberately put Harry with the Dursleys to either toughen him up or make him reliant on various other parties and that he set up the meeting with the Weasleys so that Harry would immediately be sympathetic to the "Light" side. Apparently this is easier than just making sure the kid grows up knowing that things like murder and torture on a basis of group supremecy is wrong. That, and if the Weasleys are Obviously Evil (as they often are in those sorts of fanfictions), how they faked a genuinely kind meeting for any length of time, particularly when two of the Weasleys Harry meets are eleven and ten, making it even more unlikely that they'd be able to stage a perfect meeting.
In countless fanfics, wizards will refer to London, or at least the parts of London that aren't Diagon Alley or the surrounding wizard alleys, as "Muggle London." This term never shows up in canon, and is a little nonsensical anyway since there is no named "wizard London."
The idea of characters relying on prophecies or acting like prophecies are infallible is very common in fanfics, but it goes against a point made that prophecies do not have to come true and many don't. Voldemort is the only one who believes his has to, and he's shown as being very narrow-minded in thinking this. Not to mention, only Voldemort, Dumbledore, Harry, Ron, and Hermione and, secretly, Snape are aware the prophecy exists, which is usually ignored in fanfics where everyone and their mother seems to know Harry is destined as the only one who can kill Voldemort.
It has become almost assumed that Harry is a descendant of Godric Gryffindor, usually based on the fact that he is a "true Gryffindor" who can pull Gryffindor's sword from the hat when needed, despite the canon explanation being that any Gryffindor who has a great enough need for it can pull sword from the hat (as shown when Neville Longbottom did it while confronting Voldemort). The idea of Gryffindor's legacy passing to his heir would also negate Gryffindor's populist ethos which is as much part of it as Slytherin's aristocratic fetish.
It's considered canon by many fans that there was a war between Wizards and Muggles at some point before the series began, and that said war is the reason why Wizards and Muggles live separately from each other. A common observation is that the infamous Killing Curse "Avada Kedavra" sounds suspiciously like "abracadabra", the phrase that young Muggles always say when pretending to do magic in jest—possibly suggesting that Muggles were once frequent victims of the Killing Curse, and that cautionary stories about it were passed down through the generations until the bastardized phrase became ingrained in the popular Muggle lexicon. The one point of disagreement, however, is on which side won: fans in the "Wizards won" camp point out that Wizards are a thriving community with complete autonomy from Muggles, which they presumably wouldn't have if they'd been defeated; fans in the "Muggles won" camp point out that the Wizarding government is called "the Ministry of Magic", and that it's apparently a Ministry within the British government rather than a government in its own right (possibly suggesting that it was conquered by Muggles and given autonomy after the fact).
The theory that the phrase "abracadabra" came from "Avada Kedavra" is unlikely as the Killing Curse was invented in the Early Middle Ages (6th-10th century A.D.) and the earliest known use of the phrase "abracadabra" is from the 3rd century A.D. If anything the incantation for the Killing Curse is a bastardization of the Muggle phrase.
There have been many fics involving a "wand holster," which Harry (and possibly his friends) might get, or else "dueling wands" with a certain amount of pre-loaded spells. Neither of these is so much as implied to exist in canon, though the wand holster is at least a sufficiently obvious idea that one can Hand Wave that canon just never mentioned it.
It's common among some fans to believe that Salazar Slytherin hailed from Ireland, since Word of God establishes that Godric Gryffindor was English, Helga Hufflepuff was Welsh, and Rowena Ravenclaw was Scottish. It helps that House Slytherin's heraldic color is green (a color popularly associated with Ireland), and that Salazar Slytherin was supposedly a parseltongue who could communicate with snakes (inviting comparisons to the old legend about Saint Patrick banishing all snakes from Ireland). The problem is that Slytherin's first name Salazar derives from the dictator of Portugal, that during the timeframe when Hogwarts was being made up, the colours of Ireland were likely to be sky-blue (as an English author with a liberal arts background like Rowling would know) and the idea of Saint Patrick banishing the snakes does not equal to control and use of snakes which is part of Slytherin's mystique.
Of course, neither Rowena nor Helga are names native to their birthplaces either ('Rowena' is Welsh and 'Helga' is Norse) so maybe it was popular for wizards and witches to give their children fancy foreign names. It's just that imagining Slytherin with an Oirish accent makes him a lot less intimidating.
Parselmouths are often portrayed as capable of communication with reptiles other than snakes, usually dragons. A related (and much sillier) notion is Parselscript, as in a written form of Parseltongue which is frequently found in books and/or documents written by Slytherin or in usually-Egyptian cursed tombs. This is parodied in one fic where Hermione points out that snakes can't read or write.
Some fans seem to assume that everyone gets their Hogwarts letter on their eleventh birthday, no matter what the date. Harry did, but then, it was hand-delivered by Hagrid on the last possible day to apply to Hogwarts (and, incidentally, Hagrid began sending them a few days earlier).
The idea that Hedwig (and/or magical pets in general) is some sort of magically-enhanced Familiar bonded to Harry has no particular basis in canon. Post owls would need to be more intelligent than the average bird to do their job, and it's often mentioned that Hedwig is particularly intelligent, but as far as anyone can tell that's just normal individual variation. And anyone who has ever had any kind of pet can confirm that they don't need magic to be affectionate towards their owners.
Since the Fantastic Racism concept of blood purity is a key component of the series, it’s not uncommon to see explorations of how real-world race and ethnicity interacts with it. Word of God implies that wizards don’t have the same concept of skin color/ethnicity-based race that Muggles do, so it’s pretty common to see fan art that portrays characters whose races were never explicitly mentioned as people of color. One very popular one is that James Potter was a black man, and his “unkempt black hair” was actually an Afro, a trait that his son Harry inherited.
Depicting Hermione — whose skin tone is only offhandedly mentioned once in the entire series — as a black woman is also common.
Snape. Snape. Severus Snape. DUMBLEDORE! (When Alan Rickman turned up at the Deathly Hallows Part 2 premiere in Trafalgar Square, the crowd behind him were audibly chanting "Snape, Snape, Severus Snape!")
Kreacher is sometimes called "Kreature", either by accident or to highlight Kreacher being a jerk.
Peter Pettigrew a.k.a. Wormtail is often called "rat bastard", because he betrayed the Potters to Voldemort and has the ability to shapeshift into a rat (most likely a pun at him being a snitch ("rat")).
The "Daily Prophet" is sometimes called "Daily Propaganda", "Daily Pravda" or "Daily Rubbish" for being a horrible sensationalist rag and the mouthpiece of the Ministry of Magic.
"Offense against the Dark Arts" as opposed to "Defense against the Dark Arts".
There's also a running gag about him being called or he calling himself "son of a bitch", since he's able to shapeshift into a large black dog and his mother was a very awful woman. Some fanfic authors give him the middle name "Orion", so his initials become S.O.B.
Dumbledore is occasionally called Dumbles, Dumb-as-a-door, Dumbledork, Dumbledoor, Headbastard, the Bumblebee, the white bumblebee, Bumblewhore, the Spider, the goat, the Dark Lord Dumbledore, the Light Lord, Dumb-Old-Bore, Dumb-to-the-core, Bumblemore, Bumblebore, Dumblewhore, Dumbfuck, Dumblefuck, Doubledork, Dumbledoof (German for "Dumbledumb") or "M.O.B" (abbreviation for "manipulative old bastard").
The Order of the Phoenix, a paramilitary magic organization established by the above-mentioned wizard Dumbledore to combat the wizard mentioned below, Voldemort, is sometimes called "Order of the Fried Chicken", "Order of the Hogwarts Fried Chicken", "Order of the Useless Phoenix", "Order of the Kentucky Fried Chicken", "Order of the Headless Chickens," "Order of the Roasted Turkeys", "Order of the Flaming Flamingos", "Order of the Ruptured Duck" or "Order of the Ugly Duckling" for being not proactive enough in the eyes of fanfic authors and fans against the Death Eaters.
Voldy, Voldypants, Moldywarts, Moldyshorts, Voldilocks, Voldemerde, Voldemrot, No-Nose, Snakeface, Tommy-boy, Snake-Ass, Voldemart, Tom the Dark Lord, Tom-Tom, Tomcat, Peeping Tom, The Dork Lord, Dark Putz, The Dark Wanker, Voldewerk, Voldewank, Voldewitch, Voldewhatever, Lord Druckermort, Wind of Death, The Lord of Farts, Voldemonkey, VD-Mort, The Noseless One, You-No-Clue, Heartless Bastard, Lord Voldipants, Lord Thingy, DorkLard Moldywart, Baldemort, Tommy, Pinball Wizard, Morty, Marvie, Marvin for Lord Voldemort/Tom Riddle. In Deathly Hallows, Peeves the Poltergeist actually uses the first, and Moldyshorts came from a parody sketch on All That.
Though we don't really see him blow up stuff (even though he could) some call him Tim.
Voldy-Vision: the occasional visions of Voldemort that Harry gets through his scar connection. Often plot-convenient, especially in Deathly Hallows.
Vapormort is used to refer to Voldemort when he was "less than a ghost". Quirrellmort is Quirrell possessed by Voldemort as well as pairing that soared in popularity after the release of A Very Potter Musical.
Or more specifically, the idea, eventually made canon, that Hermione would marry Ron and Harry would marry Ginny, thus integrating the whole main cast into said family.
Harrycrux - The (now confirmed) theory that Harry is an accidental Horcrux.
Ho!mione or Whoremione - The Mary Sue version of Hermione with "curves in the right places", found in badfics. There's also "Ho!rry" and Whorecrux for slutty fanfic versions of Harry.
Ginny Jo Sue - the Ginny seen in books 5, 6 and 7, after she underwent a rather dramatic change of character that led to accusations by some of her being a Mary Sue. She's also known as Sassy!Ginny or Ginny II. Anti-fans who dislike the character may refer to her as "Gin-bot," "Gin-slut," "Snogdoll" or, interestingly, "Jenny" (after a typo in a German news article).
Worthless Canon Brats - Teddy Lupin and the respective Potter, Weasley and Malfoy children who appear in the epilogue. For obvious reasons, used by people who hated the epilogue and/or the canon ships.
Moonstruck: Harry/Lupin fanfics.
Puppy Love or Wolfstar for Sirius/Lupin fanics.
Wotcher Wolfie for Lupin/Tonks.
Gin and Tonic for Ginny/Tom Riddle.
The Government Stole My Toad for Luna/Neville.
Guns 'n' Handcuffs for Harry/Draco.
Orange Crush for Harry/Ginny.
Harmony or Pumpkin Pie for Harry/Hermione, the latter based on a fanfic where they kiss and Hermione notes that Harry tastes of pumpkin pie.
Fire and Ice for Draco/Ginny.
The Good Ship for Ron/Hermione.
Lunar Harmony for Harry/Hermione/Luna.
House Sparklypoo - The fifth House where Mary Sues belong. Showed up in a one-shot Fan Web Comic and the name stuck. On Livejournal, user Pottersues came up with three more Houses for different types of Sue: Bitchiwitch (Jerk Sue), Tootsitramp (slutty Sue) and Qanonreip (Sues that ignore Canon depictions of character, and often add random elements from other fantasy series — "ooh, it's Professor Legolas, isn't he cute?").
See also the Harry Potter Summer, aka the summer of 2007, when both Deathly Hallows and the film of Order of the Phoenix were released within two weeks of each other.
Harmonians (or Harmoanians): derogatory term for Harry/Hermione shippers, anecdotally the most unhinged people/designated whipping boys in the fandom. Certainly only the more radical shippers would keep going past getting Jossed.
The stereotypical hardcore Harmonian is defined by the following traits. They believe the series was ruined because Harry and Hermione didn't end up together. They think Hermione is too good for that dumbass Ron, engendering the hatred of Ron fans everywhere. They don't think Harry is quite good enough for that perfect goddess Hermione either, but he's the closest anyone could come since he's The Hero. They believe that Harry and Hermione's love so purely transcends base lust that it's evidenced by the very lack of any evidence. It's frequently assumed that Harmonians are either misguided feminists who see themselves as Hermione (if female) or fanboys who want to get in Emma Watson's pants (if male).
Occasionally reappropriated by those, who approve of this pairing.
McGoogle/Preacher McGongle for McGonagall, Dumbly, Voldemint, Snap, Loopin, Lucien, Serious Blak etc. - all from My Immortal
The Yule Brawl for the giant fight in Book IV after the Yule Ball, for the pun.
The pimp cane: The snake-headed walking stick used by Lucius Malfoy in the movies.
It's also called Snakey.
Pink Power Granger: Hermione as seen in the third film, in which she wore a pink hoodie during the climax.
The "blood quill": The bloodletting quill (a quill, that doesn't use normal ink, but the writer's own blood as an ink-replacement. The text written with this quill also appears in form of scars on the back of the writer's hand) which Umbridge forces Harry to write "I must not tell lies" with in the fifth book, due to the lack of any canon name.
Blood Quills were never shown as illegal in Canon... though they probably were (since corporal punishment was outlawed, much to Filch's dismay).
It was confirmed that the blood quill was invented by Umbridge.
The Scottish Book: Refers to Rowling mentioning that she might publish an "encyclopaedia" of the Potter universe, including information that didn't fit in to the books. Now looks less likely with the advent of the Pottermore website.
The First Wizarding War and the Second Wizarding War: The wars against Voldemort, again due to the lack of canon names. The First Wizarding War is the war which occurred in the Back Story and ended when Voldemort failed to kill Harry as a baby. The Second Wizarding War is the war which occurs during the series, starting in Goblet of Fire and ending in Deathly Hallows. Often Wiz War 1/2.
Sometimes also known as Vold War I and Vold War II.
After the release of the final film, Professor McBadass has become quite popular for Professor McGonagall.
Slytherfen: derogatory term for fans who think the Slytherins aren't as bad as the narrative voice paints them (especially when compared to the actions of the main characters).
There's also "Snapefen" for fans who are excessively devoted to Snape and paint him as the most important and/or worthwhile character in the series.