Hombrew is a fancy word for "PSP Applications." (Yes I know it sounds like homemade coffee) Anyways with homebrew you can control your computer, use your PSP as a second monitor, use common IM clients, check your email, run emulators, you can run linux and windows 95 ( and mac ) with homebrew ! The list could go on for a couple of steps!This is QJ.net
They have lots of homebrew for many consoles. I encourage you to find an app at QJ or PSP-HACKS
and come back, then I will show you how to install an app.
Now that you have found an app (and I have refilled my glass of iced tea :)) You need to extract the .rar or .zip archive. Then dig around and check out the readme (its way easier if you learn the controls firest !) after that we have to copy the directory containing EBOOT.PBP and a bunch of other files to our memory stick in the X:>PSP>GAME directory !
Now test out your homebrew app ! If you have any problems ASK in the COMMENTS then other people can learn from your problem or mistake !
NOTE: Some apps only work on the Phat, some only on the slim, and sometime they don't work at all !