It's really hard to pick out my favorite, because I've read a lot more manga than I have watched anime. But in the case of aime, I'd probably have to say Gunslinger Girl: Il Teatrino. It is truly a work of art capable of moving me to tears.
Gunslinger Girl: Il Teatrino (the sequel to Gunslinger Girl) is well-written and full of great character development. It has great music, and great action scenes, and great animation. (Though critics would disagree with me on that last one.) This series got bad reviews from several critics, but screw them. Those guys just didn't understand the complexity of it like I did. This series touched me, and not in a naughty way. And I actually learned something from it.
I can't say much about it without going into full-on review mode, which I'd rather save for an actual review. But to keep things short, the series taught me that war is morally gray, and that often, both sides have a valid reason to fight. This series showed me how people who are labeled as "bad guys" or "terrorists" are often just like everybody else, and live unique, interesting lives like everybody else, and have redeeming qualities about them like everybody else. This series taught me that all deaths are tragic, whether it was a death on your side or not.
So yeah. It's an amazing work of art if you pay attention to it. (And don't let the change in animation style cloud your judgment!)
EDIT: Or possibly Black Lagoon could be my favorite too. It didn't teach me quite as much, but it also moved me to tears and I made memories with it when I watched with with my friends.