In T.A. 2790 Thrór and his companion Nár went to Moria to see the ancient halls of their ancestors. Despite the warnings of Nár, Thrór went through the East-gate. He was captured by Orcs under the command of Azog, who beheaded him. Azog wrote his own name in Dwarvish runes on Thrór's head and threw it outside the gate for Nár to retrieve. He let Nár go to tell all the Dwarves that an Orc ruled Moria now, and so Nár did, which started the War of the Dwarves and Orcs.
In T.A. 2799, the War came to its climax with the Battle of Azanulbizar, which was the largest battle of the war. After the battle Dain II Ironfoot, the victor, would not allow his people to reclaim it, for he had once looked through its doors, and knew that the Balrog lurked there still.
On January 15, T.A. 3019, the Fellowship of the Ring, after the loss of Gandalf, exited Moria through its eastern access, still broken. It was not heavily guarded. Prior to that, both Gandalf and Aragorn had passed through it.
Fans sometimes refer to the Great Gates as "East Gate of Moria", in contrast to the "West-Gate of Moria", the Doors of Durin. The name "East Gate" was never used by Tolkien.