I would recommend one of the 26 episode ones:
Cowboy Bebop's quite a good one, great soundtrack, interesting characters, poor bounty hunters trying to make cash kind of theme.
Trigun is ace, though it's kinda old now. Really funny, story of vash the stampede, the humanoid typhoon who seems to wreak havoc wherever he goes
Berserk is pretty cool and has a good plot going on if you don't mind hacking and slashing and blood. Story about Guts, badass guy with a huge sword, and his loyalty to a mercenary group called the band of the hawk and to their leader, griffith.
Full Metal Alchemist is worth watching too, though it's 52 eps. Two brothers try to bring their mother to life and end up with one trapped in a suit of armour and the other losing an arm and leg. They're now trying to get these things back by finding the philosopher's stone. Some funny bits and some depressing parts, particularly the episode with the chimera
There are others that most anime fans have watched such as Eva (the most depressing mecha anime in the world...ever), naruto (great ninja anime provided you skip the 100s of filler episodes) and Bleach (main character works as a death god, gets uber powerful and fights bad guys with a little help from his friends).
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You've been nothing but an angel every day of your life and now you wonder what it's like to be damned...
"This is my story. It'll go the way I want, or I'll end it here"
--Final Fantasy X