The Gift is a region to the south of the Wall under the control of the Night's Watch. It lies at the northern edge of the region known as the North. It was donated to the Night's Watch by House Stark when the order was founded thousands of years ago, in order to support the Night's Watch with food and provisions.
While cold, the Gift still contains good, arable land, far away from the wars of southern Westeros. As the Night's Watch declined over the centuries, however, increasing wildling raids over or around the Wall have results in the lands of the Gift closest to the Wall to be largely abandoned. The abandoned town of Queenscrown lies within the Gift. The closest village to the Wall within the Gift in which common peasants actually live is Mole's Town.
By definition, the Gift is ruled by the Night's Watch, and thus contains no noble Houses. Therefore no special bastard surname was ever developed for the region, because these are legally only used by the bastard children of nobles.
In the books
The Gift is officially not subject to the authority of Winterfell, and is technically not part of "The North", but is a special administrative zone directly ruled by the Night's Watch. Culturally and socially, the lightly inhabited villages of the Gift are usually seen as just an extension of the North. The Targaryen Conquest did little to change this: nominally, the Gift is part of the unified realm of the "Seven Kingdoms" under the Iron Throne, but in practice the Targaryen kings respected the strict non-interference rules of the Night's Watch, allowing the Watch to continue to functionally control the Gift without royal oversight.
The Gift originally extended for twenty-five leagues (seventy-five miles) south of the Wall. This was doubled some two centuries ago when Queen Alysanne Targaryen, the sister-wife of King Jaehaerys the Conciliator, visited the Wall on the back of her dragon, Silverwing. Impressed with the valor, bravery, and loyalty of the Night's Watch, Alysanne urged her husband to extend the Gift. He did so by adding the "New Gift" to the existing territory. The New Gift extends for an additional twenty-five leagues to the south, giving the Night's Watch control of all the lands from coast to coast for one hundred and fifty miles south of the Wall.
The declining numbers of the Night's Watch have reduced their ability to both hold the Wall and defend the villagers of the Gift from the small wildling parties that are sometimes able to bypass the Wall (by either climbing over lightly-defended areas or going around by sea). As a result the Gift is lightly-settled close to the Wall, with more farms and villages located in the New Gift close to the lands of House Umber. The most notable incursion into the Gift was by Raymun Redbeard, the King-Beyond-the-Wall of more than a century ago, who reached Long Lake before being defeated by the armies of House Stark.