Differences between PSP models
Solution 1:
The Wikipedia entry for the PSP has detailed hardware specs. Things to watch out for:
- PSP Go has no UMD drive
- PSP 1000 has no video out, the more recent models do
- PSP 1000 gaming controls aren't quite as good as the more recent models
- some accessories may only work with a specific model of PSP
- PSP hardware is region coded (represented by the last digit of the model number), as are most UMD movies and some games
Solution 2:
A bit more technical info:
PSP 1000 AKA PSP-PHAT is the first generation model of PSP. having 32 MB RAM.
PSP 2000 AKA PSP-SLIM is the second generation model of PSP, sporting 64 MB RAM.
PSP 3000 is an upgraded version of PSP 2000 with a better screen having the same amount of RAM as psp2000.
PSP E 1000 is basically PSP 3000 with some things removed (WI-Fi, TV-Out, Stereo Speakers). It has also a worse quality screen than the previous version of console.
PSP GO has the same hardware as PSP 3000 but a smaller screen and flip controller style also you cant use UMDs in this.
PSP VITA is the next generation of PSP tech specs could be found on the following link http://en.wikipedia.org/wiki/PlayStation_Vita
Solution 3:
Crisis Core is (still!) only available on UMD*, so until Square Enix rectifies that situation, you'll need to buy a PSP with a UMD drive to play this game.
As other answers have stated, PSP models 1000, 2000, 3000 and E1000 (ie. all models except the Go) have UMD drives.
If you buy a PSP new, it will either be the 3000 or the E1000, since I believe they are the models currently manufactured. Either is perfectly capable of playing all available PSP games (both UMD and digital). I find Wi-Fi useful on my PSP, so between the 3000 and the E1000 (which has no Wi-Fi), I'd recommend the 3000.
I myself have a PSP 1000 (the original model) and a PSP Go, and both are upgraded to the latest firmware. I haven't had any problems playing the latest PSP games on the PSP 1000, but I tend to buy games digitally these days and play them on the Go.
* I can't find a definitive reference for this, but if you google for "crisis core psn" you'll find plenty of recent whiney forum posts about it.