I generally like any kind of anime that has distinct, well thought out characters (or at least really cool characters, or character driven story/plot), quiet retrospective kinds, and awesome plot kinds. I guess in anime/manga speak I like seinen mostly, with a few drips of josei and some light shounen on the side.
(however, shoujo is my super secret vice, so shhhh, don't tell anyone I know irl. :x)
Some of my favorites: Mushishi, Cowboy Bebop (like you didn't see that coming), Darker Than Black (I blame Hei for being so damn sexy), Haibane Renmei, Miyazaki/Ghibli misc (Princess Mononoke being on the top), GiTS, Serei no Moribito.
Ones on my to-watch list that look quite promising: Monster, Mononoke, Gankutsuou.
Edit: Krat said this: "Anime/Manga is alot like Sci-Fi and Fantasy novels if you ask me.
The vast majority is a steaming pile of pig crap but then you come across an Enders Game or Code Geass and it all becomes worth the effort of looking the the diamond in the rough."
And I cannot agree enough. I experience much pain with that sci-fi/fantasy novel industry thing, simply due to the fact that for every 'omg did not see THAT coming' moment, there is more of the 'wtf is this? Is this. . .is this a fantasy ROMANCE novel?!' or 'Wow, another series about high fantasy that sounds exactly like that other generic martin/goodkind/brooks/tolkien ripoff of a ripoff, or hell, of Lovecraft,' which is even worse because 9/10 times they suck worse than the high fantasy ripoffs.
Anime is the exact same. Yes, there are some anime with depth. No, you are not going to find them unless you dig, and unless you know how to dig. Chances are, if you ask someone else for some 'good ' anime, they're going to recommend you something you think is crap. Hell, my most favoritist anime might be nothing more than boogers to you. That's cool for you to not like it, but do know that *my* favorite anime/type of anime (and everyone else's ) prolly isn't even a fraction of what you can find out there.
And that is why I like anime so much. It's as wide and diverse a medium as fiction books. I'm sure without the sheer amount of depth (I would insult Neal Stephenson, China Mieville, Neil Gaiman, Lovecraft, Orson Scott Card etc etc etc if I even suggested a thing), but there is *some * depth, and I don't always want Stephenson for entertainment, so anime is nice for that middle ground.
Plus sometimes I like looking at pictures. Oooh, shiny.