A dog is one of the possible pets your child can adopt in Skyrim: Hearthfire
Unlike other pets, you bring the dog home and then talk to the child who will ask if the dog can stay, if you agree than the dog becomes the house pet and stays in the house when you are not using it.
Note: If your child does not ask about adopting the dog and instead just says something like "Stupid dog", fast travel to another town then come back and try again.
You can bring home the following dogs for the kids to adopt as pets: