There is no such thing as "an Uruk-hai". Uruk-hai is a plural form. Saying "an Uruk-hai" is equivilent to saying "a Men" or "an Elves".
Uruk is the word "applied as a rule only to the great soldier-orcs that at this time issued from Mordor and Isengard" as Legalos noted above. Uruk-hai is a collective term for these great soldier orcs.
The term "goblin-men" is used in the chapter Helm's Deep by Gamling, while Merry speaks of Men with 'goblin-faces' in Flotsam and Jetsam.
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Tar-Elenion
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