Yes using the wii shop channel which is on every wii console. It uses the internet connection, either LAN using a usb LAN thingy or wireless. The games on the wii shop channel are from previous...
If you're talking about the wii internet on the wii menu then the answer is, yes you can download some games such as PAC'man and other classics. but you can't download other wii games created for the...
you either buy your games in a store, or you download them from torrent sites, which is considered illegal by most people. Assuming you mean "illegal downloading" you'd have to find a torrent site...