[Tekkaman]

Well, I am Tekka, so I'll add my two yen to the discussion. If you have WiFi/wireless internet, you can connect right out of the box with no additional bits to buy. If you don't have WiFi, you can either a) get a wireless modem so that you do have WiFi, or b) buy the ethernet adapter for the Wii which allows you to connect an ethernet cable into the console for wired internet access. (Connecting it via USB will not work, Brodyman.)

Personally, I find it to be more sensible to buy a wireless router than to buy the ethernet adapter. This way, you can avoid having to run cables from your computer to your Wii. In addition, if you have or get one, that would allow you to connect a DS to the internet as well.