Shagrat was the captain in command of the Uruks at the tower of Cirith Ungol, a watchtower that guarded a treacherous pass into Mordor.
After the discovery of the unconscious Frodo near Shelob's Lair, Shagrat and Gorbag had Frodo put into the highest room of the tower. While the two were searching through Frodo's things, a dispute erupted over the highly valuable mithril shirt. The quarrel led to a battle between their respective units in the tower, where almost all were killed, culminating in Shagrat killing the wounded Gorbag.
Shagrat then fled with Frodo's possessions and gave them to Sauron, but his later fate is unknown. The possessions were later presented by the Mouth of Sauron as supposed proof of Frodo's continued imprisonment.
In Peter Jackson's film trilogy Shagrat's role is somewhat confusing. He is portrayed as a large, Mordor Uruk and of clearly different stock along with the other orcs of the tower of Cirith Ungol from Gorbag, yet he claims Frodo's mithril shirt as his own. After the fight among the orcs however he smuggles the shirt past Sam and delivers it to the Dark Tower.
Read more about this topic: List Of Middle-earth Orcs