The Twilight Hack is the original safe, public way to enable homebrew on a Wii without hardware modification. The Twilight Hack is achieved by playing a hacked game save for The Legend of Zelda: Twilight Princess which executes a homebrew application from an external SD card. Examples of such homebrew elf files can be found on the Homebrew applications page. The Twilight Hack was created by Team Twiizers.
Twilight Hack 0.1beta1 is compatible with System Menu up to 3.3, 0.1beta2 is compatible with System Menu 3.4. The twilight hack is not and never will be compatible with System Menu 4.0. An alternative exploit, Bannerbomb, has been released and can be used instead of the Twilight Hack on all System Menu versions.