The name "Uruk-hai" has the element Uruk, which is a Black Speech word related to Orc, related to the word "Urko" in Tolkien's invented language of Quenya. The element hai means "folk", so "Uruk-hai" is "Orc-folk". A similar term is Olog-hai ("troll-folk"), used for a breed of especially strong and vicious trolls capable of surviving sunlight.
Christopher Tolkien describes "Uruks" as an anglicization of "Uruk-hai" and his father used the two terms interchangeably a number of times. While "Uruk-hai" means simply "Orc-folk", the term was reserved for the soldier orcs of Mordor and Isengard, with "snaga" ("slave") being their term for other breeds.
--Wikipedia, using this as reasoning for the word's importation to English.