Latest Custom Firmware: PRO-CFW 6.60.
PRO CFW is the most widely used Custom Firmware right now due to its great compatibility which allows you to play any game (new and old) without the need to patch it.
If you are looking for a guide on installing PRO CFW for your PSP, click here for the official guide.
If you are looking for the latest version of PRO CFW, click here for the download page.
If you are interested in trying out the Nightly Build, click here. Remember, try this at your own risk as it might brick your PSP.