A Wii Friend is a person who you have registered, either by trading Wii Numbers or by registering their e-mail address.
Registering Another Wii
To make a Wii Friend you must first obtain the other person's Wii Number. Next you must go onto the Wii Message Board, and go to your Address Book. Then, select the "Wii" option. After that you must simply type in the Wii Number of your friend. The friend will then appear in gray text. If they complete the same process with your Wii Number, then you will be confirmed as friends.
Adding E-Mail Addresses
To register an e-mail address as a Wii Friend, you must follow the procedure above. However, instead of clicking the option to register a Wii, you must select to register an e-mail address. You must then type in the e-mail address. A message will shortly be sent to the e-mail address, which they can respond to. After they respond, you can communicate with them through the Wii Message Board.