An animal friend can serve any companionship purpose - even the exotic ones we would not normally consider as pet material. Exotic pets have their own specific needs and requirements, so before choosing an exotic pet one should ask the appropriate questions.
Befriending a pig or a sheep on a farm is not unheard of. Many of the animals we consider to be livestock are quite intelligent and personable. Cows are social creatures, and horses attach themselves quickly to a consistent rider or caretaker. Homeowners with backyards can have a potbellied pig as a pet, which acts similarly to a dog and can sleep in the house with the family.
In the home, lizards, turtles, and snakes are great pets for those who are gone a great deal for their careers. They require little attention, and aside from climate control and regular food, serve a basic companionship purpose without the intrusive neediness of a dog or mood swings of a cat.
Fish are a beautiful addition to a home, and a great first step for young children in learning how to care for an animal. Fish tanks can be as small or as big as you wish. Saltwater tanks are an in-home science project for both adults and children, as they require some work initially to begin, but relatively little to maintain, as they are an enclosed ecosystem.
Birds can also be fun, although they require more human interaction than the above mentioned animals. Birds like to be spoken to, asked questions of, and sung to. They, in return, will sit on your shoulder or desk and whistle or chirp back. Birds require a clean cage and fresh food/water daily, as well as a space to adequately stretch - unless you plan to allow the bird to roam within your house, at which point a wing clip will be necessary.
Rodents can make good family pets if correct handling is enforced. They can bite, so training them to not bite can be done with treats. Some can also be shown in 4-H, similar to livestock. As a side note, all pets of the rodent variety (hamsters, rabbits, gerbils, rats, mice, and guinea pigs) have the bite tendency. It takes time and patience to reduce this sometimes painful behavior, but it can be done.
Ferrets may also make wonderful pets for someone willing to invest the time. They need several hours of play and attention daily, and can be expensive due to health issues. They are little clowns, and give their owners many laughs with their antics. Since they sleep the majority of the day, little space is needed for them, but special cages, food, and medical care may add up.
There are also species of animals that people seek as pets but are illegal in the United States due to species protection laws and concerns about diseases. It is safe to assume that any pet you find in your local pet store has been cleared by authorities and is able to be brought safely into your home for you and your family's enjoyment.