Irish: Stone of Destiny
According to the earliest texts--namely the Lebor Gabala Erenn (Book of Invasions), this Stone of Destiny was brought by Morfesa the druid from the city of Falias, when the Tuatha de Dannans invaded Ireland. It was set at Tara, where it was believed that the stone would cry out when the true king of the island stepped upon it.
I'm not sure how this would work, as the stone that is there now is a tall, upright column, which until recently was known by the locals as "Fergus' Phallus"--Fergus in this case being Fergus mac Roich, ex-king and wandering bard of Ulster.
Scottish tradition says that the Stone of Scone, upon which the kings of Scotland and later England were crowned, is part of the original Lia Fail; however, they are clearly two different types of rock.
Later historians (in the medieval sense) wrote that the stone was actually Jacob's Pillow (some sources, perhaps focusing on the shape of the actual rock, call it Jacob's Pillar), the rock upon which Jacob-Israel saw the stairway to heaven brought to Ireland by Tea and Temhair, twin daughters of Jeremiah and eponymous queens over Tara ("Temhair" is the Old Irish spelling, which we Anglicize as "Tara"). One can see this later tradition as a confusion with the original De Danann version.
Lia Fail is also the name under which Michael Penn publishes his music.