The first thing you will need is an SD card and an SD card reader connected to your PC.
It is a good idea to format your SD card before you begin; I had a number of problems with homebrew applications that were fixed after I reformatted my card. I formatted it in FAT16 (also just called FAT) on the advice of some guy on Yahoo Answers who says the Wii reads and writes faster using FAT16 than FAT32.
If you have previously used the SD card to install or attempt to install homebrew you might have a file on your SD card called boot.dol. If so, delete or rename it. The same is true if you have a folder on the card called "private."
Optionally you can also put some applications on your SD Card at this point, or you can wait until you make sure everything installs okay before you bother with that. In this guide I will choose the latter option. You can find information on installing homebrew applications to your SD card on the last step of this guide.
The method for installing homebrew is somewhat different depending on the operating system of your Wii. To find out what version of the operating system you have, go into Wii Options, click on "Wii Settings" and check the number on the upper right corner of that screen. That's your OS version. If you have 4.2 or lower you will use something called Bannerbomb. If you have 4.3, you will use Letterbomb.
If you're installing homebrew using Letterbomb for OS 4.3, click here.
If you're installing homebrew using Bannerbomb for 4.2 or lower, click here. (We're skipping steps 3 and 4, which are devoted to the Letterbomb Set Up.)