This topic comes up quite frequently (and I always seem to have the same responses

)
For feature-length animations, I'd recommend
AkiraPerfect Blue; while not for everybody (some of the material contains deeply shocking scenes of rape), the story is solid...worth watching at least once.
Ghost in the Shell (moreso based on others' recommendations than mine; I haven't watched it completely yet)
Grave of the FirefliesMononoke Hime (Princess Mononoke)
(and my favorite film of all-time, not just anime)
Kaze no tani no Nausicaa (Nausicaa of the Valley of Wind), done by the same director as
Mononoke Hime, Miyazaki Hayao.
For series, I'd recommend:
Serial Experiments: Lain -- easily my favorite series of all of them. Weird, heady, psychedelic...some of the themes explored in here have been done before, but the style of the series is wholly unique, the conclusion satisfying but thought-provoking (not to mention quite a tear-jerker, at least for me).
Cowboy Bebop -- each episode is self-contained, but builds more upon the character and stories of each of the players. Set in Earth's future, its surface plots about a bounty hunter tracking down bad guys seem at times shallow, but it's the introspection and the character development that truly shine; almost every episode left a lump in the throat at the end.
If you like fantasy role-playing games, check out
Record of Lodoss War, probably the best thing ever put on video or film in that genre until
Lord of the Rings came out this past year.
And for the fun of it,
Dominion: Tank Police -- I laughed my *ass* off in so many scenes...a genuinely hilarious short set of skits well worth checking out.