Now don't get me wrong - I don't want to come to the big showdown in the book and have the villain defeat the protagonist. No - I want to see something in the villain that makes me root for his or her ultimate redemption.
One of my favorite villains is Mayor Prentiss in Patrick Ness' Chaos Walking trilogy. Mayor Prentiss is a nasty guy - a mass murderer and an extreme misogynist - but you understand his motivation. In a world where men broadcast their "noise" (all internal thoughts) and women don't, women have a huge advantage over men. Mayor Prentiss wants to live separately from women and learn to control his noise so he can regain his political advantage. He wants Todd on his side, and so many times (especially in the second and third books), Ness makes both Todd and the reader desperately hope that Mayor Prentiss really is turning over a new leaf.
Another villain I love is Mr. Octavio Phipps from Paul Bracegirdle's The Joy of Spooking trilogy. Mr. Phipps wants nothing more than to see Spooking - the terrible town on the hideous hill - razed to the ground. And we learn during the course of the series that he has very good reasons to wish Spooking ill. But Joy, our protagonist, is Spooking's biggest fan - which puts her at odds with the devious Mr. Phipps. And although Mr. Phipps performs some despicable acts, his character experiences so much growth over the three book series, that I almost wanted to give the poor guy a hug. And there aren't many villains you can say that about.
Did you ever find yourself rooting for a villain?