Grand Theft Auto: Chinatown Wars
Scribblenauts
Super Scribblenauts
New Super Mario Bros.
Sonic Colors
The Legend of Zelda: Phantom Hourglass
The Legend of Zelda: Spirit Tracks

Limiting myself to just games that use the touch screen/other DS hardware functions:

- Kirby's Canvas Curse was the first game that showed what the touchscreen can really do for gaming.  It's also still one of the best touch-control games you'll find.

- Elite Beat Agents is one of the best rhythm games out there for any system, and it's entirely-touch based (you didn't get a DSi XL, right?).  Just be aware that it can be quite demanding on you and the hardware!

- Trace Memory is not a fantastic game in and of itself, but the developers really went all out to find ways to use nearly all (touch screen, microphone, clamshell design, etc.) of the DS' features in ways that are both logical and fairly fun.  Since you can get it for a song, it's worth picking up.

- Picross 3D.  Awesome game.

- Rhythm Heaven is similar to EBA in that it's a touch-controlled rhythm game, but it's also more complex (and at time frustrating) as a result.

- Professor Layton series, especially 3 and 1; full of fun and interesting puzzles, with great characters and an amusing storyline.  As long as you can tolerate Luke's voice acting, you'll find little to complain about with this one.

-  Moon/Dementium II: the only FPS games on the DS that work out alright.

-  Trauma Center series features some unique gameplay:  you're a surgeon, and you use the touch screen to operate on your patients.  While I prefer the Wii games over the DS ones, the tactile feel of the touch screen does add a certain something to the games.

That should get you started.