A map made by Thrór the King under the Mountain, after he had been exiled from his kingdom by the Dragon Smaug. As was traditional among the Dwarves, it was drawn with East at the top of the map, rather than North, and showed the Lonely Mountain of Erebor, the Running River that flowed from it, and the lands about. It also contained mention of a secret entrance to the Mountain, and (written in secret Moon-letters) instructions as to how it might be opened.
In his old age, Thrór set out to wander the lands of Middle-earth, but before departing he gave the heirlooms of his house to his son Thráin, including his Map. Years later, Thráin too set out into Middle-earth, seeking to recover Erebor himself, and taking with him the Map to guide his way. He was captured in the Wild by the servants of Sauron, but Sauron captured Thráin for one of the Seven Ring of Power he bore, and so overlooked the Map he carried too. This proved to be a serious oversight, because Gandalf secretly entered Sauron's lair at Dol Guldur while Thráin was a prisoner there, and managed to escape with the Map and the key that opened the secret door.
Using Thrór's Map, his grandson Thorin was able to lead a small group of Dwarves (and one Hobbit, a certain Bilbo Baggins) on his own quest to refound the ancient Dwarf-realm. Through many adventures, they were able to recover Erebor and bring about the destruction of Smaug, dealing Sauron a serious reverse.
Date of creation
According to Elrond, the map was drawn on a midsummer night that had the same moon phase as the midsummer of T.A. 2941. Astronomically, this occurs about every 19th year. Therefore, in the years between the sack of Erebor and Thrór's death, T.A. 2778 seems the most feasible date of creation.