Durin's Tower was the tower upon the Silvertine. It was built by the Dwarves in ancient times on the peak of the Silvertine, which they called Zirak-Zigil - one of the three Mountains of Moria under which lay the realm of Khazad-dum. The tower was carved out of the living rock of the mountain and stood upon a high eyrie. The Endless Stair led to the tower from the foundations of the mountain.
By the end of the Third Age, Durin's Tower had long been lost. Some Dwarves believed it had been destroyed, while others thought it was only a legend. On January 23, 3019, Gandalf and the Balrog climbed the Endless Stair to Durin's Tower. Upon the Silvertine they fought the Battle of the Peak from January 23 to 25. During the battle, the tower was destroyed and the entrance to the stair was blocked.
Source: The Two Towers: "The White Rider," p. 105-106