A fan story is a story written by fans about a favorite tv show, movie, book, or almost anything else you can think of.
Fan stories can come in many types of genres: angst, action and adventure, anime, alternate universe, self insertion, and many more. Fan stories can be about almost anything relating to the tv show or movie that the story is depicted on. The original canon can be in Japan and the fan story can be in the U.S. or France or in a made up world that the writer created. There can also be pairings in a fan story such as NaruSaku, ShikaIno, and even yaoi stories such as SasuSaku.
A fan story is also another way to describe Fan Fiction.
There are quite a few websites that are based on fan fiction such as fanlib.com or fanfiction.com
Kylewin of fanlib.com wrote an amazing Naruto time travel fan story that's over 23 chapters long!