The OP asked about anime, which is simply the Japanese word for cartoon (animation); it's used for any animation, including western. It's you who read it to mean only Japanese.
Anyway, some of my favourites:
The Vision of Escaflowne - the series, not the movie; best anime soundtrack ever.
Ghost in the Shell: Stand Alone Complex - especially 2nd GIG, which had a more cohesive story than the first season
Code Geass - though I hated how half of the battles went like "oh no, there's no way we can win; but wait, we have an ace up our sleeves: this new technology that no one has mentioned before!" Deus ex machina, anyone?
Love Hina - pretty girls and lots and lots of physical violence at the expense of the protagonist; what's not to like? Plus I get to go "ooh, I know where that is" in all the episodes set at Tokyo University
On the western side: Animaniacs and Freakazoid. Nothing will ever beat the WB/Spielberg cartoons of the 90s!