Anime:
Grave of the Fireflies: Given. If you didn't cry, you either did NOT pay any attention, or you have a void in place of your heart. One of those movies that you watch once, cry, and never watch again. Not because it was bad, but because it was TOO SAD.
5cm Per Second: Makoto Shinkai's work in general has gotten to me to tear up, but this one made me cry.
Not sure if this counts, but I did cry during the Hisui ending of the Tsukihime visual novel game. It was done in Anime style, so I guess it counts.
Cartoons:
Up!: Up! had me cry two times. It was actually the first American made animation that made me cry.
That's really it. I mean, things have made me tear up now and then, but only a few things can really make me cry.
Edit - Somebody mentioned Maus, totally forgot about that one. I read it for a History class I took, it was amazing. I don't remember crying (I know I teared up though), but it came close enough to warrant a mention.