There are essentially two different types of PSP, a hackable and an unhackable. Yours is an unhackable. However, you can run a "lite" custom firmware that runs in memory yet still is for most purposes a full custom firmware. The only downside is that when you turn off the PSP or the batteries dies/is taken out, then you'll need to run the exploit again to "turn on the CFW" again.
What you're looking for is called 6.60LME and it is available here in our Downloads section. If you try to run a full custom firmware like 6.60ME, then your PSP will brick.
Update from NeurOn. Place the "installer" and "launcher" folders into your "GAME" folder on you memory stick and execute the "installer" first. Once thats done, execute the "launcher" and it will re-boot into LCFW