One reason I love anime is the vast array of human experiences and emotions it deals with. So what are the anime moments that have touched people the most deeply? "Moving" doesn't have to mean "sad", anything that strikes strong emotion of whatever flavor qualifies! I'll start:
Number one is definitely Mugen's first death-and-rebirth scene in Samurai Champloo. It's beautifully done and captures Mugen's fierce spirit, the bleakness of Ryukyu life, and much much more. And the haunting music ("Obokuri Eeumi" by Isue Asazaki, if you were wondering) is just perfect.
Second: from the One Piece manga, Tony Tony Chopper's heart-rending backstory in the Drum Kingdom arc. This 48 year old hardass must admit to shedding a few tears while reading this... maybe because I know intimately what it's like to be an outcast among one's own kind, but not fit in anywhere else, either. The cherry-blossom snow part was affecting, too, in a happier but still bittersweet way.
Third: One Piece again, near the beginning of the Skypeia arc, when the Luffy and crew ride the Knock-Up Stream into the sky. Totally felt like I was sharing the joy and danger of those moments!
Honorable mention: Saiyuki and Saiyuki Reload manga. Full of intensely moving scenes, which are not harmed by the fact that they all happen to incredibly luscious bishonen.
OK, let's hear errone else's!