Of Azog's origins we know little; he was the chief of the Orcs of Moria, and apparently the most important Orc of the northern lands. The average lifespan of an Orc is unknown, but it is conceivable that he was one of those sent to Moria by Sauron in about T.A. 2480. He had one son that we know of, Bolg, who succeeded him after his death.
Azog entered history in the year III 2790 due to King Thrór's desire to revisit and perhaps refound the lost realm of Khazad-dûm, in whose mighty ruins Azog dwelt. Azog captured and slew Thrór, and branded his hewn head with his own name, AZOG:
When news of this reached Thrór's heir Thráin, he was greatly angered, and mustered a force of Dwarves to seek revenge on Azog. So began the War of the Dwarves and Orcs: the Dwarves hunted Azog, and many battles were fought beneath the earth. After nine years of war, the climactic Battle of Azanulbizar was fought before the gates of Moria itself.
King Thráin and his son Thorin were in that battle (it was here that Thorin gained the surname Oakenshield). Near the end of the battle, Azog himself emerged and fought with Thráin's cousin Náin and, after breaking Náin's neck, his young son Dáin. Though only 32 years old, Dáin swept the head from Azog's shoulders, and helped to win the day.
The Dwarves made no attempt to press their advantage, though, because Dáin had glimpsed Durin's Bane during his battle, and warned the Dwarves not to attempt to enter Moria. Azog's underground dominions in the north, though greatly reduced in the War, fell to his son Bolg, who held them for more than 150 years until he too met his end at the Battle of Five Armies.