It's the same as baby-sitting, but you look after a domestic animal (pet) instead of a baby.
I assume that it could work in a whole lot of ways, depending on your business model. I would choose an area where there are many pets and pet-owners, and where those owners are wealthy.
PS: It could also be somewhat different from baby-sitting in that owners might leave pets for weeks while they go on holidays, but that is usually referred to as putting your pet into a pet hotel. I guess it would be unusual to get someone to live in your home for weeks just to look after your pet. A next-door neighbour who calls in once a day to feed the animal while the owners are away might also be called a pet-sitter.