As mentioned there are several levels of 'hacking'
Almost all options require the installation of the HomeBrew Channel. That's what the Twilight Princess hack does. It enables WiiBrew. SD card required. Clear instructions given on the wiibrew.org website. Don't install the latest firmware update!!

Another variation is to install a mod chip. I won't go into details here.
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Finally the USB Gecko (and WiiRD software) is a very powerful tool that makes hacking easy.
This graphic describes it best: http://l0nk.org/geckoos.png

It makes it *too* easy IMHO. Hacks that took an hour to find on the old DS, you can find in a minute or two with the Wii (and USB Gecko).

People are trying to make 'codes' without a USB Gecko. That is possible, but it is slow and error-prone (like trying to write a program without a debugger).

The WiiRD tool forum has a lot of useful info:
http://wiird.l0nk.org/forum
(like any other public forums, expect a considerable number of newbies mixed in with people who know what they are doing)