Nearly all the members of the animal kingdom (save for a few special species) are social animals.
Finches are no exception; these birds need the companionship not only of the human owner but of other birds as well (preferably, another bird from the same species). So when you are ready to buy from a breeder or a pet shop, we highly recommend that you purchase at least two finches.
If you can, purchase two healthy male and female finches so you can also breed the birds later on. Avian offspring will easily come if the cage or aviary is constructed well and you are able to provide good lighting, adequate heat, and a healthy combination of seeds and live food items.
If you buy a single finch and place that animal in a small cage, the finch will experience stress and loneliness. You will not be able to fully enjoy the company of this exotic bird because it is pining for the companionship of its own species.
Often, finches can get used to other species of birds just so it can have someone to interact with inside the artificial habitat. Species like zebra finches are not picky with companions; as long as there are other birds in the cage or aviary, the zebra finch is relatively happy. So when you are buying a finch, always ask for at least two of each species.