Fanon has several definitions, and is a term that applies to almost every media-based or Internet fandom. As defined in the TV Tropes Wiki, fanon is "the set of assumptions 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.'"
In other words, fanon is what the fans think fills in the gaps in the original show, and deliberately avoids contradicting canon itself.
Fanon becomes especially potent when put into writing, either in essays or fanfiction, as it is then likely to be referenced thereafter by other fans. If fanon becomes especially prevelant, it can be mistaken for or misremembered as being canon by newer fans.
In Daria fandom, fanon is anything said or implied about Daria that is not strictly canon, but does not contradict canon and is therefore widely accepted by most fans.
The source of such material often has a semi-official tone to it. Before 2011, when Highland was canonically placed in Texas, most fans still located Highland in that state based on comments in an interview with the creator of Beavis and Butt-head, Mike Judge; they also assumed that Lawndale was located in a state like Maryland or Pennsylvania (usually Maryland), based on an interview with Daria creator Glenn Eichler.
At other times, the fanon is an extrapolation from what's presented. Tiffany Blum-Deckler is an Asian girl with both an Anglicised first name and a double-barrelled Jewish surname, leading to the fanon that she's the adopted child of a Jewish family. Mr. DeMartino, due to his military school background, age, and early comments on the Vietnam War, is sometimes assumed to have fought in Vietnam.
An important aspect of fanon is when those commonly held assumptions about a show are contradicted by later official material, especially if aired on TV. This is called Jossing, after Joss Whedon, the creator of Buffy the Vampire Slayer. (See the link for a full explanation.) Early Daria fanfics were sometimes Jossed, and the entire genre of Trent/Daria romances was Jossed when the show explicitly ended Daria's crush on Trent and had them both admit it wouldn't have worked between them.
Now that the series has ended, however, nothing can be so discounted; our assumptions about Daria are (sadly) safe. In fact, wide use of the alternate-universe convention within this fandom has weakened potential Jossing to the point that most stories that diverge from canon can be enjoyed on their own merits because "it all takes place in another Dariaverse." The last time "Jossing" was a risk was when Glenn Eichler gave an interview to Kara Wild in 2005-06. Mr. Eichler stated the ages of the Barksdale sisters (Helen had sometimes been assumed to be the youngest), said "Mad Dog" Morgendorffer was a drafted WW2 soldier (he'd been assumed to be a career soldier and often placed in the Korean War), and mentioning other previously unheard-of facts that hadn't made it into the show proper.
The longer the original series remains off the air, the more authority is given by fans to other fan creations, and the stronger fanon becomes in the minds of all. While this might place the acceptance of canon at some risk, the fandom does what it needs to do in order to survive and prosper.
"Fanon" in the context of Daria fanfic is any fan-created character, name or nickname, device, location, event, or concept popular enough to be used by other fans in their own works: a literal fan-canon.
Fans have given Daria the middle name Anne, Louise, or Marie, each of which has been copied by other fans in their writings; characters like Sandi and Stacy have been given full names. (None of these conventions explicitly defies canon, and could therefore be considered fanon in the original sense) Background characters have also been given names, such as Tori Jericho for "Popular Girl" or Scarlett for an unnamed goth. Even stories that have completely different characterisations or roles for these 'backgrounders' will use the names "Tori" and "Scarlett"; this makes them instantly recognisable to many older fans, and a writer may feel free to write about them as they would about canon characters, without needing to explain many details. (See also shared characters.)