They both have different kind of functions and are similarly priced with a difference of around £20.
The Nintendo DS supports a duel screen, one of them being touch sensitive, and a stylus in which to write with on the touch screen with. Disk again is very useful and makes it just that little bit better. The games on the DS however have less quality graphics than on the PSP and the games are mainly more 2D than on the PSP which has more 3D games with better graphics.
The PSP is a portable Playstation, capable of storing all kinds of media on the device. These include videos, photos and music files. These are very useful and in this way the PSP outranks the Nintendo DS. The PSP has high definition graphics that are 3D. This makes game play more realistic. By having a memory card to keep data on, UMD videos or game disks are able to make multiple saves which can be very useful.
One problem with some of the PSP games is that some games have to be cut down and shortened which can make game play a bit more dull and you might not enjoy the game as you would to its full extent.
On the whole the PSP is better than the DS in most ways and although it does not have special features like a touch screen, it has better quality games with better content on them. The PSP is reliable and useful and is very good all round and I guarantee that it will entertain you very well in every way possible. The DS is also very small and will give you entertainment to a certain extent but it is not in the same league as the PSP.