From Bulbapedia, the community-driven Pokémon encyclopedia.
Fanon is an unconfirmed belief held by fans of a series. Such beliefs may be based in canon (the things that have been confirmed), or they may not be.
In the Pokémon franchise, fanon is not restricted to the anime, but most instances of fanon are related to it. For instance, it is canon that Delia Ketchum is Ash's mother. This is said in no uncertain terms; it is an indisputable fact. However, any conjecture of Ash's father is purely fanon, as the only mention of such a man in canon is how long it took him to get to Viridian City, and that he is supposedly a great Trainer. Everything else is speculation.
Most Shippings are fanon, although some are canon. For instance, Norman and Caroline are a confirmed couple, whereas, say, Jessie and James (RocketShipping) are not in the anime, though they are in the manga series The Electric Tale of Pikachu, which has its own canon. A shipping like PearlShipping is pure fanon. Shippings can also be half-canon; that is to say one of the characters in the shipping has a crush on the other, such as AmourShipping, where Serena has a canonical crush on Ash.
Fanon is often a way for fans to express their own beliefs about the games or anime and speculate about future episodes or games. For example, many people hoped for a remake of Gold and Silver, even before Nintendo officially announced the games.