I have had both, basically I got a DS to pass the time until the PSP debut, so considering both, heres a comparison:
Nintendo DS:
Not only are the Dual Screens (One a Touch Screen) innovative, but most of the DS games take full advantage of these features, creating gameplay that can't be reproduced on almost any other system, even consoles. The stylus/thumb nubule has been utilized in many ways: Analog Control (Quite succesfully for FPShooters and Driving/Flying), Puzzle and Writing Recognition, hell, even Mock-Surgery. The second screen also adds a much appreciated status screen for some games that would (On other systems) require you to pause and sift through menus to get the same effect. An integrated microphone is also utilized (Though not much) but seems to do a good job of word recognition.
The DS has great game support, with many different DS titles to choose from, Action, Adventure, RPG's, Driving/Simulation (This is where the touchscreen shines), even Survival Horror (Yes, Resident Evil for the DS)and Sniping (Yes, Silent Scope), and as of late, DS has been coming out with these "Brain Training" type games, which I have personally tried and have nothing but good things to say about. Then you have the backwards compatibility, which basically adds countless Gameboy Advance titles availiable (Can you say Final Fantasy? YES!!!)
PROS:
* Innovative Dual/Touch screen design
* Games actually take advantage of DS's design to make unique yet fun gameplay
* Quite effective stereo sound
* Clamshell design protects screens from everyday abuse
* Backwards compatible to Advance games
* Good battery life (4-9 Hours, DS lite even longer)
* Low price (Regular DS $100 used, DS lite $130 new)
CONS:
* Underpowered when compared to the PSP
* Cartidges are chip based which limits size of games, and therefore content
Sony PSP:
The first thing I thought when I opened up my new PSP was "SEXY!!!!!!". This thing is simply beautiful, not just the sleek design, but the crystal-clear huge screen (For a portable) shows vibrant colors and lines. The PSP is wonderfully engineered with a Beautiful wide screen at an unbelievable 4.3" (Huge for a portable), stereo speakers, and authentic Playstation-style controls, even an analog tab (Although not the most accurate analog I've used).
Whereas the DS handles like a PDA when you power it up, the PSP handles more like an actual PC or a media device, with a menu that lets you play games (or any other type of media off of the memory stick). The games themselves sport graphics that are a cut above the DS, but not quite up to PS2 standards, lets just call it the PS1.5 graphically. And lets not forget the Movies that are avaliable, pretty much DVD quality, but IMO the movies would seem better off for someone who travels/flys alot. Not knocking it, just find all those "Home theatre" mounts for the PSP with the speakers on them a little weird when I can just watch the same movie on my much larger TV.
I have to admit this now, Sony really dropped the ball when they launched this badboy. The PSP would have smeared the DS off the bat, but sony made 3 major mistakes that really slowed down the PSP's progress:
A) They force-fed you to buy the PSP via a "Value Pack" that pretty much included worthless stuff to milk more money out of the consumers, effectively raising the price from $175 to $250. That turned off alot of people.
B) They had almost NO game/movie support at launch, and, for the first 4 months thereafter.
C) They sold ALOT of defective units (Some with dead pixels on screens, some with dead "X" buttons)
PROS:
* Sleek and beautiful features
* Large high definition screen
* Great stereo sound
* Authentic Playstation controls with a true analog stick
* UMD format created large capacity for games
* Large capacity Flash Media sticks gave almost endless storage space
* UMD Movies are DVD quality
CONS:
* Screen unprotected and quite fragile
* So-so battery life (3-4 hours maximum)
* Moderately high priced still due to the "Value Pack" at $250
Unfortunately I had to sell both my DS and PSP when I came upon hard times a year+ back, but recently I decided to get one again, after careful thought and consideration, I went with the DS due to the current Game Support and remembering the fun I had playing it vs the still somewhat fun but mostly "Eyecandy" feeling I got from my PSP.
I hope this helps.