Homebrew is the method of using unofficial software (and sometimes hardware) to a game console. Only use this tag when referring to hardware or software modifications; DO NOT use about pirating software (i.e. free download and/or distribution of licensed games or other paid software).
Homebrewing is the method of running games or software onto a game console that was not licensed by the manufacturer. One can usually homebrew via software modding only, however some methods require use of modchips or other homebrew kits that mean disassembling your console and using a soldering iron.
Homebrewing can also be achieved through errors in a game (i.e. a "Twilight Hack" was made due to an error found in the Wii version of The Legend of Zelda: Twilight Princess), however these only work as long as the errors remain unfixed.
Some independent games have been developed with the use of homebrewed consoles, the most popular being games developed by fans of the console. Unfortunately, several games have been shut down due to copyright restrictions.