An heirloom tomato is an open-pollinated, non-hybrid heirloom cultivar of tomato. They are classified as: family heirlooms, commercial heirlooms, mystery heirlooms, or created heirlooms. They usually tast much better, have a shorter shelf life and less disease resistance than hybrids.
The seeds are what make an heirloom tomato an heirloom tomato. They are passed down from season to season, taken by the farmers from the tomato plants that produced the best fruit. This process allows farmers to select for certain desirable traits like juiciness, size, shape, or color. Heirloom tomatoes are also often open-pollinated, which means that they are pollinated naturally, by birds, insects, wind, or human hands. No weird science-y genetic modification here.