Someone who is really into something (or someone). The word's origins are from a shortened form of "fanatic"; "fan" implies the dedication, but doesn't have so many negative implications. (On the other hand, the word "fanatic" originally merely meant "one who is inspired by religious zeal"; given the heights to which some fans take their devotion, that definition is not entirely inaccurate.)
One can be a fan of something (or someone) for a multitude of reasons. Perhaps it is incredibly good, and you are in awe of it - or at least consider it one of the good 10% of what's out there. Perhaps you respect it for its historic importance. Perhaps just part of it is good, but it's a part you value. Perhaps it is bad, but in an entertaining or historic way.
If a work is to be truly remembered Exty Years from Now, it must have fans of some sort. If no one cares enough to watch it, no one will remember it when it's out of print: this can easily lead to Older Than They Think. If a fandom once existed but has died out, Pop-Cultural Osmosis and "Weird Al" Effect are likely to have occurred in the meantime.
An active fandom can keep the memory of a work alive after it's gone out of print. (Just look at the Sailor Moon fandom around here.)
The plural, genderless form of "fan" is "fen", for when 'Fans' just won't cut it.