Things you will need:
A Wii console
An SD Card (formatted in FAT16 or FAT32)
A computer with Internet Access
An SD Card reader for your computer
Things you should know:
If you don’t know what homebrew is, read my article.
The Wii was not designed by Nintendo to support homebrew. There is no guarantee that using homebrew software will not harm your Wii. About.com does not take any responsibility for any problems arising out of installing homebrew. Proceed at your own risk.
It is also possible that installing homebrew could void your warranty.
Future Wii updates to the Wii may kill your Homebrew Channel (or even brick your Wii), so you shouldn't update your system after installing homebrew. To prevent Nintendo from automatically updating your system, turn off WiiConnect24 (go into Options, then Wii Settings and you'll find WiiConnect24 on page 2). You can also learn how to prevent new games from attempting to update your system here.
It is a good idea to read the Wiibrew FAQ before proceeding.
Do not buy a kit to install homebrew on your Wii. All homebrew tools can be found for free on the internet; these kits simply repackage these free tools.