You should definitely watch animes from Masaaki Yuasa if you want something different (Cat Soup was already mentionned):
- Mind Game
- Kemonozume
- Kaiba
- The Tatami Galaxy

And these animes were already mentioned by mrcrispy83 but those are some of my favorites as well:
- Serial Experiments Lain
- Haibane Renmei
- Texhnolyze
- Kino's Journey
- Mushishi

Animes from Satoshi Kon are also a must watch in my opinion:
- Perfect Blue
- Paranoia Agents
- Millenium Actress
...

*cracks knuckles*
Okay, I know I'm going to repeat what some others have put but here's my list:

Trigun
Outlaw Star
Lupin III (films or series)
Cowboy Bebop
FLCL
Hellsing Ultimate
Saiyuki (original and film, not Reload or Reload Gunlock)
Claymore
Supernatural (the anime is actually really good)
Slayers (everything series, films and OAVs)
Tekkaman Blade
Samurai Champloo
Full Metal Panic: Fumoffu
Gurren Tengen Lagann
Darkside Blues
Vampire Hunter D/Vampire Hunter D: Bloodlust
Monster
Evangelion (films and the new OAV only)
Elfen Lied
Ikki Tousen (original, Dragon Destiny and Great Guardians)
Giant Robo
Castle in the Sky
Nausicaa
Howl's Moving Castle
D. Gray-Man
My Life on Arcadia/Space Captain Harlock
Black Lagoon
Elf Princess Rane
High School of the Dead
Melancholy of Haruhi Suzumiya
Grave of the Fireflies
Ghost in the Shell
Akira
Baoh
Bubblegum Crisis
Dirty Pair
Cutie Honey
Paprika
Tokyo Godfathers
Darker Than Black
Ninja Scroll
Gungrave
Boogiepop Phantom
Astroboy (original or 80s version)
Gunslinger Girl
R.O.D. OAV
Perfect Blue
Kite
Ranma 1/2 (mostly the early seasons and the oavs)
Dominion/New Dominion Tank Police
Ah My Goddess!
Tenchi Muyo/Tenchi Universe
Metropolis

notice all the GunGrave? yeah its that good!

Listen to this guy, hes hip.

Are you suggesting that Bible Black is not a serious anime?

Doesnt crunchyroll require paid subscription now? (read there is a week delay for new eps for free users...)

What kind of show are you looking for? Asking for anime recommendations is no different from asking for book or game recommendations, your going to get a lot of recommendations but they may not be what you are looking for.

People have already hit on most of my recommendations (Ghost in the Shell, Neon Gen, Case Closed, ect), but if you are looking for a good tragedy, Bokurano Ours. That show deconstructs the whole kids saving the world with giant robot, and the anime is actually the lighter version of the manga.

For lighthearted comedy, Hetalia (although the fandom is a bit on the crazy side :/).

Scifi: ./hack sign, Blue Gender

Movies, Anything by Studio Ghibli (Kiki's Delivery Service, Howl's Moving Castle, Castle in the Sky, etc).

Twelve Kingdoms
Trigun
Hellsing

If you don't mind watching movies, I would suggest Children who Chase Lost Voices from Deep Below. Absolutely brilliant. Does Ghibli better than Ghibli does (these days).

I'd suggest others, but chances are they aren't available outside of Japan and as Crunchy Roll does the standard "fuck off if you're in Japan" routine, I can't even check that to see if they are available to you via that.

Most of the bases are covered. Crackle.com has a number of things including Blood+, and Bubblegum crisis tokyo 2040 for free. Too bad they are in English.

One thing I didn't see yet I liked was Project arms. The production value was uneven, being real rough the first few episodes then getting progressively better (along with a sudden more mainstream look in season 2 ) but its worth checking out once the staples have been exhausted.

Some of my personal favorites are Trigun, Lain, X, princes mononoke, and Iria. Ninja Scroll is amazing although very, very graphic. Spriggan and Ghost in the Shell are visually stimulating.

I think Saiyuki, Saiyuki Reloaded, Samurai X are pretty good with lot of character development, in a good way. Gungrave for bromance and amazing plot. Samurai X fighting scenes are realistic and well-detailed. It's not magic or fantasy stuff like Bleach, Dragon Ball. Jigokou Shojou (Hell Girl) also quite fun, scary and cheesy at time. It's bit serious and mature.

My favorite shows in order (I have watched 150+ anime, so these have a special place in my heart lol)

1.Fate Zero
2.Steins;Gate
3.Cowboy Bebop
4.Great Teacher Onizuka
5.Welcome To The NHK
6. Ano Hi Mita Hana no Namae wo Bokutachi wa Mada Shiranai
7. The Tatami Galaxy was previously mentioned, it's a bit weird to start with, but it's a masterpiece. Utterly brilliant ending.
8. Usagi Drop

Some good beginner anime:
Durarara
Baccano
Black Lagoon
Samurai Champloo
Bakuman

Rape, Futanari, demon shit, enemas....I can go on.
No, just very fucking weird...like La Blue Girl.

Besides many of the mentioned ones:
Utawarerumono
NOIR
Armitage III
El Cazador de la Bruja
Murder Princess
Tower of Druaga series
Macross
Macross Frontier
Brave Story
Key the Metal Idol
Berserk
The SoulTaker
Royal Space Force - The Wings of Honneamise
Escaflowne movie and series
Bakumatsu Kikansetsu Irohanihoheto