Well, most of what I'd recommend has already been suggested, so all I can do is put in an extra recommendation for stuff like Ouran Highschool Host Club (It really is an excellent series, I should re-watch it again sometime), Steins:Gate (Again, a great series, I was practically on the edge of my seat while waiting for the next episode to be released each week), and Spice & Wolf (A decent series, at least. I like it, but I feel the second season focussed too heavily on economics, rather than the good balance the first season had).
A good chunk of the series that Imperator_DK has listed are worth at least checking out as well, I' say.
Oh, and for all of these, I'd recommd watching subbed rather than dubbed.
Especially if you decide to watch Higurashi No Naku Koro Ni.
That animé has possibly the worst English dub I have ever heard.