What with the media-saturated modern world we live in, it is unsurprising that once in while someone becomes an excessively hardcore fan of a work of fiction, such as the notorious "trekkie" fans of the "Star Trek" franchise. Fan Art is when fans take a comercial series into their own hands and create new content based on it for their own satisfaction.
It is illigal—in any country that recognises intellectual property rights—to comercialize this fan-produced material, but fans—-ranging from amateur artists to full-time professional artists, produce it anyway out of passion. Tight-knit communities have even formed around popular subjects, such as The Risembool Rangers fanclub devoted to anime star Vic Mignogna, simply because such is a sincere common interest.
Fan Art can take any creative format, from pencil sketchs in the margines of one's homework, to masterpieces of the digital canvas, to character costumes called "cosplay", to writtern works called "fan fiction", even to original musical compositions. (I don't know the chic term for that.) They are all great ways of starting out on the long and perrilous journey to becoming a better artist; inspiration is easier to come by when it's a subject you're already excited about, and there happens to be a built-in community of likewise-excited people willing to critique your work, and who probably want you to critique theirs.