Fan fictions (fanfic for short) are fictional stories written by fans. For example, you're a fan of Card Captor Sakura, you can write a fanfic about Sakura getting marry with Syaoran or whatever.

Keep in mind that fan fictions are not offical stories, they don't really happen in the series. And if they do, it's pure coincidence.

First of all, you need to have some knowlegde of the series you're writing for. For example, you can't write a fanfic about Superman and say he has a batmobile. Once you have all the knowlegde you need, then you can just let your imagination flow.