I agree it has been covered several times already but I might as well take this, what I am about to prattle on about also applies to the TA-088v3+ PSP2000s ("unhackable" slims) and is not far off what those with "hackable" slims (PSP2000s which have a hackable motherboard) and the PSP1000 (indeed I flashed several this way). Those with a "hackable" PSP in the conventional sense also have access to a pandora battery and associated software which means it is hackable at any time not just with the method I am about detail.
If you are past official firmware 5.03 you are out of luck, trade it with someone who has a lower firmware or trade it for a hackable PSP. No word of any useful exploits for later firmwares at this time (ok there was a proof of concept but how long it will take to become workable is up in the air).
Before or at 5.03 is different as you have access to chickHEN- an exploit than runs from the system menu of the 5.03 firmware (it exists in some earlier versions but was only compiled/made for 5.03 and it is not like it is hard to update to 5.03).
With chickHEN you boot into the so called "HEN" firmware. This alone is not enough to play games but it does give you homebrew which in includes custom firmware installers.
At this point hackable slims and PSP1000s can just flash a custom firmware using their method of choice (hellcats for me although others have other methods).
Do not repeat
do not do this to a PSP3000 or a unhackable slim unless you want to brick it.
Reasoning: the unhackable slim and PSP3000 check their firmware so if you overwrite it with a custom firmware then it will check then see it is not proper and refuse to go any further.
All is not lost however as people tweaked the custom firmwares to be able to run on top of chickHEN (actually it patches the system menu in memory but same that is more or less a conceptual difference) and still have the original firmware in place on the flash.
That is to say you run chickhen, you run the CFW enabler and then you have your custom firmware on a PSP3000.
Naturally if you turn off the PSP you clear the ram and have to run chickHEN and the load up the hacked firmware all over again so most people stick it in standby instead (you can also do it for a hackable PSP by nobody really bothers to do such things as you can just flash those).
ChickHEN is slightly different in terms of what you have to do to get it working depending on the version of the PSP you have (PSP1000 is different to the PSPslim and PSPslim and lite aka 3000).
Here is chickHEN for PSP3000:
http://pspslimhacks.com/guide-to-chickhen-...sp-3000-ta88v3/
Here is a CFW enabler guide for once you have done that:
http://pspslimhacks.com/custom-firmware-en...-3-30-tutorial/
As the guide will say chickHEN is 100% and sometimes it will just crash the PSP instead, reboot and try again as it will eventually run (if it was not obvious you can tell by going into the system menu and checking the firmware version).