In The Return of the King, Shagrat and Gorbag fight over a shiny shirt. The garment is Frodo’s mithril shirt, a lightweight coat of mail made from the rare metal mithril.
Frodo received the shirt from Bilbo, who had acquired it during the quest of Erebor. Bilbo later gave it to Frodo before the Fellowship left Rivendell. Although it looks like an ordinary bright shirt in the film, mithril is exceptionally strong and valuable, making it a prized piece of treasure.
After Shelob attacks Frodo, the Orcs at the Tower of Cirith Ungol discover the shirt among his belongings. Shagrat wants to claim it, while Gorbag’s side also sees its value. Their argument turns into a violent fight, helping create the chaos that allows Sam to rescue Frodo.
A common mix-up is calling the item armor or a vest. Those descriptions are broadly understandable, but the film’s dialogue calls it a “shirt”; more specifically, it is a mithril coat of mail. The same item later becomes evidence used by the Mouth of Sauron to convince Aragorn that Frodo has been captured.