A villain is a Bad Guy. Villains generally do Bad Things to Good People, and occasionally to Bad People, too. Compare with antagonist; contrast with hero.
Depending on their degree of evil, villains may commit a variety of crimes from the petty to the disgusting. A few follow some sort of code of honour, while others are completely without morals, and some are intelligent – perhaps to the point of being geniuses – while others are terminally stupid.
Some Mary Sues have villains as Lust Objects. Being in love with them usually involves downplaying their evil deeds, claiming they are "just misunderstood," and woobifying them in order to "reform" them. If they're not pretty to begin with, the Sue may give them a makeover a la her rose-colored glasses and/or go on at length about inner beauty and how nobody else is wise enough to see it. Very occasionally, the Sue's pure love is literally all it takes to turn them from the path of darkness.
Villains frequently subject to this treatment include Judge Frollo, Jareth, Loki, the Phantom of the Opera, Sephiroth, Sweeney Todd, and Lord Voldemort. The true poster boy for victims of reformer Sues, however, is probably Severus Snape. While not necessarily a villain per se, he is mean, nasty, and ugly; the few positive traits he expresses are rooted in envy and selfishness. Yet, the fandom can't seem to resist putting him in a sympathetic light, even going so far as to recast his greasy hair as "naturally shiny." When something like that happens, you know you're dealing with an OC who's not playing with a full deck.