In addition to what Gaffer has mentioned here's a few more to check out...
Bomberman
Worms 2
Animal Crossing
Mario & Sonic At The Olympic Games
Mario vs Donkey Kong 2
Trauma Center 2
Bleach: The Blade of Fate
The Simpsons
Just a few that I always seem to find myself playing.
But yeah, there's far too many good games to play on the DS. Loads of Brain Training style games if they float your boat (they make me feel like a thicko) and most of the big name games get a DS version too, NFS, Fifa, PES etc etc. Having said that I tend to find games that were designed with the touch screen in mind are better than games which are just ported over to make more money.
When I had a PSP I used to quaff at the simplicity of the DS and how it's graphics werent as good but after playing one that all changed. The games are infinitely more fun to play compared to the PSP, which is what it's all about really. It's not hard to see why they're so popular.
Just buy one, you wont regret it