Fanfictions, or fanfics, are stories based off of someone else's story using the original writer's characters and/or settings, usually written by fans. Because fanfictions are not yours (since you are using someone else's character/story/setting), fanficions get published very rarely. Fanfictions are also rarely get commissioned or authorized by the original author.
Because of this, the fanfiction writers usually place a disclaimer in which they state they do not own any of the characters, settings or any plots, unless it is said differently (FCs are a good example).
FCs are fan characters, more specifically, characters based on a fandom. Imagine if you created a character in the anime Fullmetal Alchemist. That character would be considered an FC.
FCs and OCs (Original Characters) get mixed up quite a lot. The main difference is that OCs are completely your characters, in a universe that is created by you and you alone. FCs are usually created based on a fandom, whether that is Digimon, Hetalia, Bleach, Naruto, or Pokémon.
Fanfictions don't have to be based off an anime. Like noted before, it can be based on any sort of fandom, whether it's based on books like Harry Potter, Warrior Cats or Twilight or video games like Pokémon and Sonic or cartoons like Phineas and Ferb or movies. Fanfiction is fanfiction, as long as it's based on something that isn't yours.
If you still aren't sure if your story is a fanfiction or not, you can contact a chat mod or an admin. If you have a question, you can ask these people as well.