An Individual Form means that the USPS will only forward mail addressed to that name. Keep in mind - if you receive mail under multiple names (perhaps your maiden name or a nickname), you'll want to file an Individual form for each name. Learn how to file another form here.
A Family Form lets the USPS know that you want all mail for individuals with a certain last name to be forwarded. If anyone shares the same last name as you and will not be relocating to the new address, you'll want to file an Individual form instead for only the people who are moving.
A Business Form is for authorized users to forward mail on behalf of their business.