Current homebrew status: Working!
The Zelda: Twilight Princess Exploit (Twilight Hack)
Requires: The Legend of Zelda: Twilight Princess (Wii)
bushing, tmbinc and segher were able to modify a save game for Zelda: Twilight Princess that would make the Wii crash and were then able to inject their own code. No hardware modification is needed. Firmware version 3.3 attempted to block use of this hack, but failed in implementation. The newest version is compatible with all Wii firmware versions up to and including 3.3.
Trucha Signing (Signing bug)
Requires: Varies. For homebrew enablement, a drivechip and a Wii below firmware version 3.3.
This method of signing content for the Wii exploits a bug in the implementation of the RSA algorithm used in some of the Wii's software. It is simple to create a fake a signature for any content, and so installing and using modified applications, game discs (with a Drivechip), authentication information has been quite simple. This bug has been patched in some versions of IOS, but since most versions are still unpatched, homebrew applications continue to be able to utilize this bug. However, because some versions of IOS have been patched, fully-updated Wiis are no longer able to run fake-signed discs, disabling a commercial product based off of the signing bug, Datel's Freeloader.
Requires: Homebrew enablement via Trucha Signing or the Twilight Hack
While not a hack in itself, the Homebrew Channel is the first homebrew loading interface that does not require any other software or hardware to run. Installation, however, requires either the Twilight Hack or the ability to run fakesigned discs. It currently works on all Wii firmware versions available up to this writing (currently 3.3). The installation of the channel relies on the Signing bug, but since its installation is done via homebrew, this is not likely to become an issue.
Wii System Updates
System Menu 3.3 & Updated IOS30, 31
This update blocked old versions of the Twilight Hack. The newest version contains a workaround, and retains compatibility with all Wii firmware versions. IOS30 and 31 have fixes for the Signing bug backported from IOS37. Since the System Menu uses IOS30, fake-signed discs have been blocked.
This iteration of IOS contained a fixed signature checking routine--a fix for the Signing bug. Although installed on many Wiis, this version of IOS has not been found active in any way that would disable any hacks.