Fanon is a fact or ongoing situation in Fan fiction stories related to a television program, book or movie that has been used so much by fan writers that it has been more or less established as having happened in the fictional world, but it has not actually been established as having happened on the show, book or movie itself.
The word is a play on "fan canon".
For example, it is "canon" that Donna lied to Congress about her diary (The West Wing), but it has become "fanon" that she lied about her diary to protect Josh (because she had written in it about his PTSD, which-- if anyone knew about it-- could damage his career). Since we were never told WHY she lied about her diary, any reason the fan base comes up with is "fanon," even if it is a pretty darn good explanation.¹
¹ Reference: [1]
Other fanons include:
Fanon is sometimes used prejoratively by purists to refer to such explanations as faulty or illogical given the nature of a story, or 'common lore' copied amongst fans (especially webpage proliferation) that actually contradicts a simpler explanation that was even alluded to in canon.
