DS. I like it just because it's so creative like everything else Nintendo makes. It's got two screens, one of them touch-sensitive, and that in itself would be enough for me. But it's also got a microphone and two ways to connect wirelessly. And download play, so you can play games with a friend even if only your friend has a copy. Also, I just like the style of Nintendo's games. They're always a blast to play, and never get too serious, though they are quite challenging. Oh, and ever play Spectrobes? You can dig up minerals and fossils by using your stylus like a drill. How awesome is that? Plus, that touch screen allows for much more precise control than a D-pad or even a thumbstick. I just like the DS completely.
IF you look past the touch screen, the graphics are much better on the psp. The ds is usually for women or little kids. It's like the Wii vs the 360 when it comes to graphics but the ds has more of a variety of games and is kind of more fun to play.
Honestly I think it all depends of the variety of games you prefer and the features. Price is an issue for alot of people, but if your really enthousiastic about it, you will try to get one or earn one somehow.