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Kxploit is a tool used to "install" homebrew applications on a PSP Memory Stick Duo. It was released for PSP v.1.50 following the discovery of flaws in the firmware that allowed the running of unsigned code. Kxploit is relitively easy to use once you have the proper EBOOT.PBP file for your homebrew app.
Kxploit received a lot of media attention after it was released. There was an earlier exploit for the flaws called Swaploit but the process required two Memory Sticks and a swap which could cause damage to the PSP if it wasn't done right.