Heritage Breeds are farm animals that have been around a long time, and in general are not used by large scale agriculture. These breeds are dwindling due to commercial unavailabilty or viability within CAFOs, but they are vital to insuring the survival of farming in the future.
For example: modern turkeys that are used on most meat farms and sold in most hatcheries have been bred to have such large breasts that the males can not longer mount the females to mate naturally: they must be artificially inseminated. Many of the most commonly sold chicken breeds on the market no longer care to hatch and raise their own young -- quite simply, the desire has been bred out of them. Some larger animals have lost some of their natural foraging and mothering instincts, along with natural disease resistance. Many pigs on large farms are being born with poor leg structure, because the breeding sows don't need to walk or even turn around in their cubicles to gestate, and no one is noticing that their legs are weak and being passed on to their young. Holsteins have been bred with overactive pituitary glands which stimulate exorbitant milk production that is results in milk laced with similarly raised amounts of growth hormone -- making more milk than an average family could ever drink in a day.
For these and many other reasons, a lot of people think its important to assure the survival of the "old" breeds which may not be super-producers but tend to be more disease resistant and better suited to life on small farms or homesteads. Smaller cows such as Jerseys and Guernseys are easier to manage and produce milk in quantities that are better suited to family use. Pigs that know how to forage are better suited to pasture life and may feed themselves for free, especially if you have great stand of oaks for them to rummage through. Baby chicks that are reared by their mamas grow up to be good mothers, too, eliminating the need to buy incubators and monitor hatching. Life on a farm, even a small one, is a lot of work: why not choose animals that help out and simplify matters wherever possible?
Even scientists are getting in on the action. Check out this NY Times article about a heritage breed sperm bank: http://www.nytimes.com/2010/01/06/dining/06frozen.html?partner=rss&emc=rss