Ultimately, Canon is much smaller than the people who throw the term around like to think it is. Canon is limited to that which has actually been described in the source material. Especially in groups of writers, it boils down to what the writers specifically need to worry about for the purposes of the ongoing plot.
Fanon, also known as 'Headcanon' among fanfic writers and Roleplay crowds, is the set of theories based on that material which, while they generally seem to be the "obvious" or "only" interpretation of canonical fact, are not actually part of the canon. Occasionally, the explanation seems good enough to just be "common sense." The salient point to remember is that when someone shouts, "That episode was terrible because it violates canon!", they are very often totally incorrect.
Fanon fills in holes that the writers may have deliberately left in order to have fodder for later stories. In addition to arising from a point of vagueness in the canon, Fanon can come into existence as a fact gained from a popular but non-canonicalsource, or taken from a different Adaptation. Because many fans mistake their own Fanon for actual Canon, they tend to get riled up when a new fact is introduced which does not literally contradict anything canonical, but invalidates what were formerly the most obvious assumptions. Many examples of Retcon and Continuity Drift that are imagined to be violations of Canon really only explicitly contradict Fanon.
Popular subjects of Fanon include character backstories, full names of characters with No Name Given, what characters actually do for a living, and Shipping — a whole other world of its own.
Since many creators in the aftermarket series universe are fans, Fanon often shows up there, and if those creators in turn start writing for the actual show, Fanon may actually becomeCanon. Alternatively, you just have Memetic Mutation within the fandom.
Fanon often also refers to the body of information provided by otherwise-official sources. Television and movie scripts are a continuing source of fanon material — Captain James Kirk, for example, had a middle initial ("T.")...but his actual middle name ("Tiberius") was originally revealed in an episode of the Trekanimated series; since that show's canonicity is debatable, it was considered "fanon" until revealed canonically in the 6th Star Trek movie. Note that this usage blurs the line between fanon and deuterocanon, though.
Warning: Fanon and accusations of Fanon are a classic Internet Backdraft, with the accusation commonly leveled by fans that have a different interpretation of the material — even when their theory is just as vulnerable to Schrödinger's Gun.
Compare Broad Strokes, where the events of a story are referenced in passing without taking everything said and done as having "officially" happened. If the fanon was repeatedly hinted at by writers until it became fanon, but never actually confirmed in canon, it's Writer Induced Fanon. See also Fandom Specific Plot. Not to be confused with this Fanon or the Pope's robe. Fanon is frequently based off Fanfic Fuel.
Many fans think Simba and Nala had a son, named Kopa, who was either then killed by Zira, or otherwise died, or is at least gone somewhere awaiting a fanfic plot that re-unites him with his lost family. A confusing example, as Kopa was a character in the semi- or non-canonical storybook The Lion King: Six New Adventures published between the 2 movies, but was then never mentioned in the 2nd film. This has been speculated to be due to grief or something, but it's more likely that he was just retconned out of existence. It's thought that he, and not Kiara, was the cub pictured at the end of the original movie. It's also due to the cub's coloration at the end of the first movie (resembling Simba's at the beginning and all the lionesses are a lighter color).
It's required in fanfics that if Kiara and Kovu have a son, he be named Tanabi.
Considering the other Lion King movies aren't officially in the Disney Animated Canon (which doesn't necessarily mean anything to the Lion King universe), some fans just dismiss Kiara's existence completely and replace her with Kopa.
Nala was meant to have a younger brother named Mheetu but he was taken out of the final product. Fans often include him in fan-works, either as her unseen brother or her dead sibling.
Due to the fact the writing says he has green eyes and black fur while the art makes him look like Mufasa, many fans have taken the third option and combined the two elements; the most common combination is his artwork build but with the colors of the text, though his eyes will occasionally look like his youngest son (Scar). His mate, Uru, often has dark brown fur with a patch on her forehead.
Nala being Scar's daughter is a very common theory. That'd make Nala and Simba related but it's a pretty well-known fanon that they're cousins..
Similarly Kovu is often considered Scar's son by many, even if the movie goes out of its way that he is not. Vitani and Nuka are almost across-the-chart considered Scar's daughter, though there's a sum of fans who believe Vitani is Nala's daughter with Scar. It's less common to believe she's Nala's and Simba's child, with her getting certain genes from both her parents.
There's also the matter of Tiana being outbid on the restaurant. It has generally become accepted amongst fans that the "other buyer" story was orchestrated by the mill's owners, either with them outright lying or with them getting someone to outbid Tiana's offer. While this does explain why Tiana conveniently is outbid the very day she tells the owners she's ready to purchase the place (and why no "other bidder" gets angry when Tiana has Louis scare them into selling), there's no official explanation for it in-movie.
The Sword in the Stone. Many fans were so saddened by the heartbroken girl squirrel that several fanfics have emerged in which Merlin turns her into a human so she and Arthur can be together. Popular fan names include Hazel and Gwen. These stories are accepted by many as legitimate Canon just because they want so much for the poor little thing to have a happy ending.
The fandom has it that (SPOILERED for Rule 34) Woody's pull-string is a sort of erogenous zone, since he lacks the necessary parts for sex. And by extension, Jessie as well, as she has a pull-string too.
Also, unless you ship Buzz/Woody or Woody/Dolly or Woody/Jessie, the figurine of Bo Peep was bought by Bonnie or her mother and is reunited with Woody.
A similar and not always exclusive theory is that Andy inherited Woody from his father, who is not seen in the films and is assumed to have left or died when Andy was very young. This would explain both why Andy has a toy modeled after a show from the 1950s, and why he's so attached to it.
In Cars fan fiction, the name of the agency Finn and Holley work for is called C.H.R.O.M.E, which is derived from the video game adaption, but the agency's name is never actually stated in the film itself.
Rise Of The Guardians had the issue of Jack's sister's name. Since she was voiced by the same actress who voiced Pippa (one of Jamie's friends) but never named in the actual film, most people thought that Pippa was Jack's sister instead. Since it's now been clarified, the general name used is Emma, although some fic writers will substitute with their own preference.
Lord Shen, the Big Bad of Kung Fu Panda 2, once had a backstory that involved him being neglected by his parents due to being a sickly albino and left in the care of the Soothsayer, who acted as his nanny. This was ultimately dropped in the final film, but it's rare to find a Shen fanfic that doesn't incorporate it anyway. This is somewhat justified, as the canon interactions between Shen and the Soothsayer do support this, and there's nothing in the film that really contradicts it either.
Even if you don't ship Elsa and Anna from Frozen, a large portion of the fanbase views Elsa as being LGBT (typically lesbian) due to her not having an canon male love interest and that X-Men has taught a generation of film-goers that "a secret power that you are born with but are forced to hide but that you then learn to emrace, making you an happier person in the process" = Gay.
Elsa quickly reached Memetic Sex God levels and this naturally occurred. She's typically depicted as an aggressive lover, at tamest a rather dominant partner.
Due to the fact she appears in a cameo, and Anna resembles her to controversial levels, Rapunzel is often seen as being related to the sisters.
Anna having powers, typically earth or fire, is common.
There's a frequently cited belief that all the Pixar films take place in one universe.
Vocaloid: The only things officially canon are the characters' names, appearances, and voices. Usually. (Some have a couple more minor things, like age and height, while others only have a name and a voice.) Personalities, backstories, relationships, and some characters are pure fanon.
Many fans believe that Garfield and Jim Davis's lesser-known second strip, U.S. Acres, take place in the same universe. While this is true on the Garfield and Friends side, it's never been confirmed or debunked in the strips.
The Ordial Plane is a concept that turns up very frequently in Planescape fan work, based on the assumption that the Astral and Ethereal Planes should have a third counterpart in accordance to the Rule of Three and which would complete the circle between the Inner, Outer, and Material Planes.
Warhammer Fantasy deliberately leaves a lot up in the air concerning some of its mysteries, but a great deal of fanon proclaims definitive answers to these questions. One common example is the nature of the Bretonnian goddess: many fans declare that she is merely a deception by the Wood Elves to be holy writ, but this is only one of several possibilities vaguely hinted at by the actual canon.
The above applies to Warhammer's sister game Warhammer 40,000 as well. There are fans who insist that the Eldar created the Tau Ethereals, based on some vague hints in the Xenology book, which had a decidedly Unreliable Narrator (and the narrator himself dismissed the idea out of hand.)
Hamlet is a breeding ground for these, due partly to centuries of theatrical interpretation and partly to Everyone Is Jesus in Purgatory induced by high school English classes. Thus, for instance, many people take it for granted that Hamlet is genuinely mad instead of faking it.
Umineko no Naku Koro ni is a case where Fanon is actually encouraged by the author. Ryūkishi wants the readers to have their own ideas and reach their own conclusions about the plot and the characters. Very few answers about the mysteries are revealed explicitly; but most readers end up acknowledging a certain conclusion (namely that Beatrice, Kanon and Shannon are one and the same) even if the story never outright states it. The manga tends to be a bit more explicit on some aspects though.
In Neopets, a popular fanmade Neopian Times piece ("Poor Dr_Death") managed to define everything pertaining to the owners of the pound/adoption center. Most notably, the anonymous Uni was given a name, and nobody has found cause to dispute Dr_Death's characterization as a lovable Deadpan Snarker. (At least, not until his official appearance suddenly became muchLighter and Softer with the rest of the website, but that's another issue.)
And that The Other Guy is the older, slightly saner brother that the Critic goes to when he needs to lick his wounds.
Again with them, any fic that takes place when they were younger calls them Doug and Guy. For common sense purposes really, it'd be silly for their parents to call them "Critic" and "Ask That Guy".
Its become Ascended Fanon that they're brothers, as Ask That Guy recently referred to Rob as his brother.
Spurred by Ask That Guy's love of his pipe and Doug's penchant for blowjob jokes, Critic being really good at oral has become almost a meme in fics.
Also that Dr. Insano's son is named "SOI" (Son Of Insano). And he goes to school.
After Kickassia was finished, it's usually accepted that the Critic was exiled to a hotel room and the others had fun on his tab. What tends to happen next is, unfortunately, wishful thinking.
Humanized versions of the Happy Tree Friends characters almost always follow a certain design set by fanon. For example, Flaky is always a petite woman with messy, hip-length hair coated in what is probably supposed to be dandruff.
Red vs. Blue has a lot of these. According to an inordinate number of fans...
Wash and CT had a romantic relationship, despite only interacting twice in the show. (In fairness, both times implied they were closer than most Freelancers, but he doesn't even react when she turns traitor... or when the Freelancers attack Charon Industries to capture or kill her. Alternately, Grif/Simmons. While Tucker does make a remark about them being in love, their relationship in the show is pretty much just ordinary Heterosexual Life-Partners.
Crunchbite was reincarnated as Junior. There's some evidence for this in the series, but it's far from explicitly said.
Epsilon is metastable/rampant. Metastability in the Halo universe is, to borrow a phrase from the forums, a big f***ing deal, involving some very obvious and extreme stages. Epsilon hasn't really exhibited any of these (except maybe anger) and it's likely he's not even capable of it (being a fragment, not a full AI), yet it's such a common fan theory that it's often stated as fact, even on This Very Wiki.
Carolina's nickname (often given to her by York) is Lina. She's never called anything but Carolina in the show, though, even by York. Alternately, her real name is Carol and she almost always is shown using the gravity hammer, despite using it just twice in the series—pistols or melee are much more part of her fighting style, yet fans have latched onto the grav hammer instead.
Sarge was previously an ODST. It is mentioned that Sarge jumped out of ships "during the war", so it's possible, but again, it's far from confirmed.
Character appearances have been pretty well cemented by Luke McKay's fan art, leading most people to believe Caboose is a blonde, Grif and Simmons have brown hair, Tex has red hair, Church has black hair and stubble, etc. While Church's appearance has sort of been confirmed (the Director indeed looks like an older, even grumpier Luke McKay Church), some of the others are definitely not correct (Tucker is implied to be black, unlike Luke Mc Kay's white depiction, and Tex is a blonde... or at least the original Allison was).
Welcome to Night Vale: Partly as a result of the series suddenly going very viral after a long slow burn, there is a Fanon interpretation of Cecil's appearance that many people mistake for Canon—including living tattoos, tentacles, purple eyes, blond hair, and glasses. Canonically, everything we know about Cecil's physical appearance comes from Kevin's non-description of his picture during The Sandstom.
RWBY: Many fics about Ren and Nora have them platonically sleeping together. This helps to ramp up sexual tension if the fic is romantic, but it's also an extrapolation of the very close partnership they have in canon.