I own both systems, and I love both of them. Though I'm becoming more fond of my PSP since its become my portable PS2.
The DS has a great solid library and some truly great AAA titles. Battery life is great, its very portable. Most of the great games are first party, but there's no doubt that Nintendo makes great games. But there are other great games, like Castlevania, Phoenix Wright, Hotel Dusk: Room 215, and others.
The PSP is also another great system. Great LCD screen, some really impressive looking titles. It takes a hit in the battery life, but that's not much of an issue for me. The PSP lacked great games when it first launched, but its getting some great games now and in the future. Sure some of the games are just ports, but they're pretty good ports (ie. Burnout, Monster Hunter Freedom, Disgaea!).
If you're one of those people who likes playing games online (like me), the PSP is the way to go. I dislike how the online wi-fi system works for DS games (but it doesnt stop me from hating Mario Kart DS, Metroid Prime Hunters, or Pokemon).
Its hard to say which system is better, only you can decide that. Go check out the game library, go look at the system in person, see what systems your friends own. Multiplayer is great with handhelds.