Fan art is original artwork made by fans of a particular franchise, such as Batman, Star Wars, etc. The art may be in any number of media: drawing, cartoons, painting, sculture, collage and photomontage, digital remix, etc.
Fan art often focuses on a particular character or motif with which the fan is obsessed or preoccupied, for instance Captain Kirk. Much of it explores the characters or plotlines in an unusual medium or style, remote from the original. For example, Darth Vader might find himself in an impressionist garden; Princess Leia might be dressed as a hula dancer.
If you google “Star Wars fan art” you will get some sense of the range of changes fan artists ring on the mainstream versions of the media. Some of them will be consistent with the vision of the films or the graphic novels; others will be as far afield as combining characters with indigenous motifs, Bollywood style, etc.
Like other fan media, fan art sometimes eroticizes characters and situations in interesting and unexpected ways. Queering straight characters is very common, as in Slash. Some artists are interested in mixing characters or motifs from one franchise with those of another franchise, for instance having Kermit the Frog from The Muppets sit and fish with Yoda from Star Wars, or Princess Leia (from Star Wars) kiss Captain Kirk (from Star Trek).