Homebrew is basically "home brewed" programs to make the PSP do things that Sony never intended thanks to holes in the security, ranging from stuff like accessing an IRC channel and instant messaging to variations of games. Some of the stuff out there is flaky at best for stability (kinda like Windows ME on computers) and the potential is there to possibly damage your PSP. This is true especially with the use of "downgraders" which are meant to take your PSP to an earlier firmware version which can run the homebrew.
Other bits of the homebrew scene are illegal- namely emulating other consoles. Technically it isn't the emulator itself that is illegal but rather the use of the game ROMs if you don't have an original copy of the game (say, an original Super Mario Bros cartridge for the NES as an example).
Personally, I would recommend staying out of the homebrew scene and use your PSP as Sony originally intended.