There was a great interview I once read with Willem Dafoe (which I'm sure I'm going to paraphrase horribly) where he was asked how he felt about playing "good" characters compared to "evil" characters. Dafoe said something like "It doesn't matter. Everybody thinks they're righteous." And that's really stuck with me.
I definitely agree. To me the best villains are fully realized characters who stir mixed emotions in you. You want to defeat them, but when the moment finally arises, you feel some kind of sadness. This sadness can come from a variety of different reasons - perhaps because they are wasted potential or because you have thoroughly enjoyed the conflict that that character presented or they have just gained your sympathy over the course of the story. They should make you ask questions about yourself or about the protagonist of the story.