These two models are both very popular in the market at the moment, for people wanting to play handheld games, on the go.
Both have very good quality graphics, and neat gameplay, but have a few differences. This is what you should look at when looking to buy.
1. The Price. The Nintendo DS is far cheaper than a PSP. With a PSP you have to buy a memory card, to save games on, or buy a "giga pack" costing £215 rrp. This adds to the cost. A Nintendo DS can be got for around £90, and offers as good quality gaming.
2. What to use it for. The main thing that the PSP has over its rivals is its ability to do more than just play games. It can access the internet (with a wireless network) and play movies, music and pictures, either from disk or transferred from the computer. If you don't feel you would use these features, DO NOT invest in a PSP, as you are paying for features you do not need.
3. The Games. A PSP has a slightly larger variety of games, with most multi- format games that you would see on an X-BOX or PS2. It also has a few of its own games, which are designed for it solely. On the other hand, the Nintendo DS has what you would expect from Nintendo. Plenty of fun, play as you like games, which are good on the go, but not great if you just want to play at home, whereas teh PSP has games such as Pro Evo 5 and fifa 06.
4. The portability. If you are only going to use your machine at home, as you don't own another games console, then look for a DS. The new version is slightly more portable than a PSP, and the folding screen makes it safer to carry in your pocket (although you can buy cases for PSP etc.)
Basically, if you want a Handheld console, look hard at your use. The PSP may be beautiful, but it may have features which you are paying for, but not using. Happy Buying!