In J.R.R. Tolkien's world of The Lord of the Rings, the Uruk-Hai were first bred by Sauron in TA 2475, and appeared when he took over Ithilien and Osgiliath. By the time of The War of the Ring the Uruk-Hai were used as soldiers by both Sauron and Saruman. They were used so widely because unlike other breeds of orc they were tall, straight limbed and would not weaken in sunlight like other orcs. They were slanty eyed and had black skin, and rightly considered themselves above other orcs.
In the Black Speech of Mordor Uruk-Hai means Orc-Race. The Uruks of Mordor were distinguishable by The Red Eye of Mordor emblazoned on their shield, while the Uruks from Isengard used by Saruman were marked by the white hand of Saruman, marked on their face and on their shields.
Gorbag, Shagrat, and Ugluk were definitely Uruk-Hai, while Azog and Blog may have been. The spies in Bree and the chief's men were probably part Uruk, if not all.
They captured Merry and Pippin at Amon Hen, but most of that group were slain by the men of Rohan. It was the Uruk-Hai that attacked Helms Deep, and defeated by the men of Rohan, the men of the Hills and the Ents, and were probably used in the final battle in Mordor.