The main anime I watch at this point is Detective Conan aka Case closed.
Wow, some good tips so far. I should probably add that I’m 22, so I probably won’t dig any of the animes that are designed more for a teenage audience. I don’t know if any of the ones that have been recommended are like that, but I’ll be checking out as many as I can. Will acquire the first few episodes and see if I like them 
Mystery mentioned “Monster” which I haven’t read, but I’ll recommend it just because Naoki Urasawa is a complete genius. I finished his series (24 volumes) “20th Century Boys” which seriously ranks among my favorite books of the last few years. I can’t wait to get into Monster. Unfortunately 20th Century Boys isn’t available in English yet to my knowledge, although a live-action movie of it is planned for next year.
For animation, check out anything from Studio4C, but especially Mindgame and Tekkon Kinkreet. None of this is really typical otaku-style anime, it’s a bit more arty and highbrow, but I dig it.
<edit>
Here’s more on 20th Century Boys. If and when it comes out in English definitely check it out:
Evangelion is all I have to say.
Oh and Death Note.
Koba
For anyone that liked Cowboy Bebop (and you should), I would highly recommend Samurai Champloo.
Second the suggestion for Planetes.
Same for GITS:SAC (Ghost in the Shell: Stand Alone Complex, 2nd GIG and Solid State Society). The original Ghost in the Shell movie is a classic too, but I wouldn’t recommend Innocence, unless you really like humanist philosophical debates.
Anything by Satoshi Kon: Paranoia Agent, Perfect Blue, Paprika
Same for Studio Ghibli, especially Howl’s Moving Castle, Nausicaa of the Valley of the Wind and Castle in the Sky
If you can look past the overused mecha theme: Fafner of the Azure, for the character driven plot.
For something that will make you want to beat up the main character but the ending will make up for it: School Days
Paranormal / Suspense / Thriller: Higurashi no Naku Koro ni (When They Cry). When the first episode starts with someone beating people to death with a baseball bat, what’s not to love.
Normal people with psychological troubles: Welcome to the NHK, does not feature any of the stereotypical anime characters, plot device, …
Fantasy adventure: Scrapped Princess
Sheer bloody fun: The Melancholy of Haruhi Suzumiya (watch in the broadcasted order, NOT the chronological order)
Meta meta meta anime: Lucky Star (I wouldn’t recommend for mainstream viewers)
Thumbs up for Monster too.
Oh, and Death Note, both the anime and live action movies (Light is such a magnificent bastard in the live action).
I’m probably forgetting a whole lot more.
It’d be easier if you tell us what you like, I can shoot suggestions for pretty much any style except sports-related anime.
Martin
Welcome to the NHK is wierd…
Pretty funny though!
I’m in the anime society here at Edinburgh Uni and I only go to watch “Welcome to the NHK” and “Death Note”. Didn’t know there was a live action version of “Death Note”!
Looking it up now.
Koba
akira
steamboy
blue submarine no 6
golden boy series (all 6 episode)
kite
md geist
cowboy bebop
for reading Yokohama Kaidashi Kikou
and BLAME!.