Enter Wii Hacking

The best and most reliable place to go to for information about the Wii code creation and methods of use is http://wiird.l0nk.org/forum/index.php

But I'll explain some of the basics - the stuff I know I can describe accurately and without much risk of misunderstandings. Everyone who wishes to help with the code creation process needs to read the official instructions, and it couldn't hurt to even if you're just a gamer, but here's something to start you off. Just so everyone knows, all you need to purchase in order to do most HB on the Wii is an SD card (preferably one that does not exceed 2 Gigs, because there can be complications with some applications if you exceed that amount), a way to hook your SD card up to a computer, and the console itself.

In order to do anything Home Brew on your Wii, you obviously need a way to boot up the programs you want to use. First, I'll explain what's been the most popular method for a long time now. What I'm talking about is the 'Twilight Hack', and it requires you to at least temporarily have Legend Of Zelda: Twilight Princess. You can rent it, borrow it from a friend, or own a copy, it doesn't matter. You need a modified save game for this to work, so if you already have a game save, store it somewhere else (not sure if there can be more than 1 on the SD and still show up).

You can get the Twilight Hack save file from http://wiibrew.org/wiki/Twilight_Hack
WiiBrew is also one of the best places to find Home Brew applications/games/etc, and that's one play you can find the code apps I'll explain in a sec.

Follow the instructions that are either included or on the site (they have a whole tutorial) so as to set everything up. There are other ways to do homebrew, but the best way to enable those ways is to start with the TH.

Now, granted, you could just use the TH method to load all your apps, but you can only do one at a time this way, and that's a lot of juggling, frustration, and unneeded waiting. Instead, you'll probably be best of installing the Home Brew Channel, which can be found on WiiBrew as well. It's their pride and joy. To get the HBC, you'll need to use the TH method (or something else) in order to run an installation app that will place the HBC in with the other channels on the wii main menu (that you see when you turn on your console). Once that's done, you can place things in the right places, in the way that instructions for each app specify.

Now for the good part. Code-related applications usually come in two different forms, for people who are using the codes and not making them. There are Code Manager (aka CM) programs (which are usually called exactly Code Manager, or something close), which allow you to choose which codes for which games you want to use (and you have to put the codes for one game in one .txt file, and stick that in a location specified by the CM you're using) and save them to a Cheat format (basically, converts them to a unique format that the next programs can use), and then there are Code Activators (that's just what I'm choosing to call them, the official name is probably something else but I don't know what).

Code activating programs can do different things. There's one on WiiBrew called Ocarina, which is only for activating the codes and starting the game, no bells or whistles. Then there's Gecko, and it's various versions and a couple variations. Gecko is very usefull for both code users and code creators, because it has features that are required for some codes and aren't currently a part of Ocarina, and because it has debug, testing, and other features for code creators.

That's the free stuff. The stuff required to make the codes is a different story. I don't know all the details, you'll have to look into the things on the WiiRD website. But I think there is at least one program out there for code creation that doesn't require the hardware they sell. In any case, I don't really know much of anything about the Wii attachment required for making codes, so I'll just leave it at that.

Hopefully all that helps, and hopefully I'm not forgetting some key thing I intended to put in there. Those links are fresh, I went to the sites and copied the URL, and they are the best places for their specific uses. Again, codes so far are only or mostly on WiiRD's code database and forum. 
Good luck, and let me know if there's anything specific you want to know that I didn't mention, and I'll see what I can do.