The Nintendo DS is my handheld of choice. I have played the PSP and I own a DS, I just don't consider the PSP very fun to be honest. Yes, the graphics are nice, and quite amazing for a handheld, but controlling the games feels awkward and not very smooth at all, especially when you have to use the make-shift analog stick. The DS just seems so much more smooth, although the graphics are far from amazing, it's just so much more fun. The touchscreen is a nice twist, opens up a whole new world of gaming. The games are fun, and most of them have a high replay value in my opinion, and replay value means more to me than graphics anyday. "Yes! This game looks great, but I beat it yesterday and probally won't ever play it again! Yay for wasting $50!"
Also, the DS is region free, meaning you can import and play games from japan or anywhere else in the world without a modchip or any additional software. I don't know if you can do the same with the PSP, but thats a good advantage for DS owners who want those games that never hit the US.
Anyway, I would recommend the DS to anyone, but I would wait for the Nintendo DS Lite to hit the US, you can buy them for $200 online but when they hit stores i'm sure they will be much cheaper (and smaller, sexier and has a brighter screen!).