They are first seen in The Fellowship fo the Ring when the wizard Saruman creates them as a more advanced version of the Orc. He sends a group of them out to find the Hobbits, who are along with the Fellowship, to find The One Ring and bring it back. They take Merry and Pippin to Saruman, as they believe they have the ring.
They are next seen in many different parts of The Two Towers, and are the most prominent enemy that the heroes fight in the story. As a large group of 10,000 strong Uruk-hai, march to Helm's Deep to destroy the surviving humans of the land of Rohan.
In Return of the King, the Uruk-hai aren't around as much, as most were killed in the battle for Helm's Deep. With the only moment of the Uruk-hai's appearance is when Sam is fighting them in the tower of Cirith Ungol to save Frodo. It is unexplained how the Uruk-hai got to Cirith Ungol. Though, some say that the surviving Uruk-hai from Helm's Deep and Isengard made there way to Mordor.
The Uruk-hai either appears or is referenced in all Lord of the Rings games.