I personally don't like the games that could/would have decent graphics on a TV-screen console (i.e. Wii) because the DS screen is so tiny. Such as Zelda, Metroid, Final Fantasy, etc. I'm sure they're fun if you're a fan of the series, but I like 'em much better when I can see what the heck is going on!
Super Mario is really fun on the DS, though, as is Yoshi's Island. Super Princess Peach is supposed to be quite fun though I didn't enjoy it too much personally. My personal favorites are Tetris (lots of new variations on the DS!) and Elite Beat Agents - that one is really cool, uses the touch screen and dual screen elements just wonderfully!
I also hear Cooking Mama and Trauma Center are excellent, I've only played them on the Wii but I can see where the touch screen controls would work well for their format.
If you like educational games, the My Word Coach games are fun! I just got My Spanish Coach and I cannot put it down!! It teaches you through very quick lessons followed by timed games using the touch screen (word search, flash cards, concentration, etc) and are very addictive. Sounds dry but there's lots to unlock and the games make me keep coming back over and over. It's really fun but I also feel like I'm learning something. I have DEMANDED My French Coach for Xmas and I am hoping they come out with lots of other languages!
Animal Crossing is also good but gets a little tedious after a while. Just got MySims for the Wii and it is a huge improvement - I don't know if it is as good on the DS but the Wii version is hard to put down!