if you have things bought on the shop and then your Wii breaks, when you send it in, they transfer everything you bought to the new console so when you get your new one, it will have all you shop downloads on it (i think. they might just give you enough points to redownload them or something)
If you have a MyNintendo account, it keeps track of all the things you have registered.
You Wii, Wii games, VC Games, etc..
You can connect your MyNintendo account to your Shop Channel account and it will keep track of what you have downloaded.