In order to play PlayStation 1 games on your PSP, you must be running firmware version 3.00 or above. PSOne games for the PSP can be downloaded using a PS3 via the PlayStation Store or you can use the PlayStation Store for PC. You will a memory stick duo of size 512MB or above in order to download games onto your PSP.
Please note that PS1 game discs cannot fit into the PSP’s UMD slot. Trying will either damage your disk or the PSP. This should be obvious since the PS1 disc is much larger than a UMD disc.