The more widespread piece of fanon in Harry Potter is the version Hogwarts uniform that includes skirt/trousers, shirts and house-specific ties, with robes on top (which are also sometimes lined with their house colours). This comes from the movies - the canon uniform consists of plain black robes, with no house markings.
This should be obvious by 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.
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, until JKR revealed that her full first name was actually Ginevra.
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.
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 was 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 had sweaters with their correct initials, the joke is that George didn't know the rest of their names. And he also apparently thinks Percy's name is 'Prefect'.
They also usually finish each other's sentences in fanfics, which happened less than a handful of times in Canon.
In canon Molly had two brothers (possibly twins?) 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.
The idea that Tonks and Charlie, who due to match likely went to school together, were very close - if not childhood friends - and had a relationship that didn't work but ended amiable out is very common, always with the inclusion of some kind of affectionate nickname on Tonks' part. You'd think this would've be 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.
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.
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!
"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.
There's also "Ancient and Noble Houses", and their magical Lord rings. The only time a character talks about them, it's Sirius Black referring to his family, and he was mocking their pureblood ideals.
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 OC 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 was in Slytherin (from the sorting ceremony in Philosopher's Stone), many fans 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.
In the first and third books, Gryffindor gets House Cup points for its Quidditch performance. This has lead to the common assumption that a House's Quidditch points (10 points for scoring a goal, 150 for catching the Golden Snitch) are simply added to the House points (5 points for answering a question right, etc), but the books don't confirm this.
Though the third book leads some credence to this since in this one it's not enough to just win the game, Harry has to wait until his team is in the lead by a specific margin, and it sounds like the win of the cup also secured the house cup for Gryffindor.
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 it's 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 you actually meet Mrs. Cole. 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 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. Also popular is the theory that she was a champion Quidditch player (usually 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.
Another near-universal belief is that Susan Bones is a redhead with (after puberty, at least) an impossibly voluptuous figure. Her hair color is never mentioned in the books, but she does have red hair in the first film, in which she was played by Chris Columbus's daughter.
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 (or at greatly enjoys) chocolate. This probably comes fromThe Shoebox Project, a fic that inspired quite a lot of common Marauder tropes. In it, chocolate is randomly Remus' Trademark Favorite Food. Everyone copied it, and it caught on.
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.
The idea that 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.
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.)
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'.
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.
Magical society discriminates against females, which is 100% fanon. Though they do acknowledge Amelia Bones as the exception that makes/breaks the rule.
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.
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 Asians. This has no merit, seeing as Cho Chang was shown to be very popular.
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 maimed 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.
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.
In next-gen fanfics, Albus Potter, Rose Weasley and Scorpius Malfoy usually form "The Silver Trio," 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.
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 never called him anything but "Ron" even when she was furious at him, and the only person who ever did call him that was 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.
Voldemort being unable to love due to being conceived under the affects 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.
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.
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.
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. 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).
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.
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.
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.
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.
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!" Awesome.)
Kreacher is sometimes called "Kreature", either by accident or to highlight Kreacher being an asshole (he was brainwashed severely by Sirius' utterly insane and racist mom).
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" or "Daily Pravda" 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 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 Roasted Turkeys" or "Order of the Flaming Flamingos" 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 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.
Voldy-Vision: the occasional visions of Voldemort 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.
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 never been mentioned in the books).
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).
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.