I'm only going to mention ones that haven't been listed yet. Also, my summaries suck, so check out the titles with Wikipedia or Anime News Network for more info:
Akagi: Akagi Shigeru, the genius who descended into the darkness, enters a Mahjong parlor one night and ends up playing a game for a man whose life is riding on the game. If Akagi loses, the man will die. Worse, the people Akagi is playing against are the yakuza, and Akagi has never played Mahjong before in his life. So begins Akagi's story, which follows Akagi through various high-stakes games of Mahjong, in which Akagi bets money he could never make working a regular job, as well as his life.
Crest of the Stars: A space opera with two people from two different worlds in a universe at war. Very interesting, nice art, and amusing characters. Has a follow-up called Banner of the Stars.
G Gundam: It's Dragonball Z with giant robots. The pilots actually fight inside their mobile suits, in special rubber suits that tell the mobile suit what to do. A very MANLY anime, but not incredibly serious.
Maria-sama ga Miteru: Shojo series about an all-girl academy which has a "sister" system where upperclassmen (upperclasswomen?) pick lowerclassmen to become their soeur (French for sister), someone they will look after through the schoolyear. The characters look good and act even better. It may seem hard to get into, but as the series progresses, the amount of comedy goes way up, but so does the seriousness. Also, you might learn a little French, lol. J'aime regarder les soeurs.
Mobile Suit Gundam (0079, the original): IT'S THE ORIGINAL GUNDAM! 'Nuff said. (Or if you need more said, then how about wonderful Char and Amuro fighting without even knowing each other, the wonderful plot and characters, and wondering just what will happen?)
Marmalade Boy: Light-hearted shojo focusing on Miki and Yuu, whose families switched partners. Each of their parents divorce, and Miki's mother marries Yuu's father, while Yuu's mother marries Miki's father. Together, the six of them move into a big house, and Miki and Yuu start living as brother and sister. Kind of. Soon, romantic feelings emerge, promising hijinks and big problems.
Dance till Tomorrow: An adult series that follows Suekichi Terayama, a college student who will inherit his grandfather's inheritance of 450 million yen if he can graduate from college and start a career. The big hurdle in this challenge is Aya, a mysterious woman that jumps into Suekichi's life, for better or for worse. Though it has graphic sex scenes (18+ only), the sex scenes are realistic and don't seem there just to get you excited (meaning that it resembles real life; people have sex in real life, you know?) Also, the series is VERY funny. It's better to buy it for the comedy than the sex (which sometimes features invisible penises, lol).
Yotsuba&!: Incredibly funny series from the creator of Azumanga Daioh. It follows Yotsuba, a little girl who is beyond weird. This is a girl that doesn't know what an air conditioner is and is scared of a balloon that is meant to scare birds away. It's a must-read for fans of comedy.