Azog is mentioned in passing by Gandalf in the first chapter of The Hobbit, "An Unexpected Party." Gandalf and the dwarves recount the history of the dwarves under the Lonely Mountain to explain their quest to Bilbo. Part of their tale includes what happened after the desolation of Smaug, when the dwarves had to flee their home. Thorin's grandfather and father both escaped through a secret door and survived; only later Thror, Thorin's grandfather, was "killed [...] in the mines of Moria by Azog the Goblin" (24).
Later in chapter seventeen, "The Clouds Burst," Bolg of the North arrives, leading a fierce goblin army to challenge the dwarves. Bolg, "whose father [Dain] slew in Moria" leads the attack and is out for vengeance against the Dwarves (251). Coincidentally, Bolg's father is Azog, former king of the Goblins that controlled Moria until driven out and killed by Dain.
Even though Azog never personally makes an appearance in The Hobbit, Tolkien uses his character to provide significant back-story to the novel, providing further insight into the long conflicted history between the dwarves and goblins.