Kyrie Eleison is a term that is used in the Latin Mass of the Catholic Church. However, I've been told that the phrase is entirely Greek and not Latin at all!
So is the phrase Latin or Greek? Is it transliterated from Greek and therefore Latin?
It is simply Greek that has been written in Roman letters.
The city of Rome was essentially bilingual from around the time of Caesar Augustus until at least the third or fourth century A.D.: the people spoke mostly vulgar Latin or common (Koine) Greek. Greek was the more common language among the poor, who formed the majority of the ranks of the Church at first. The Roman liturgy, therefore, was originally composed in Greek, and was only translated into Latin in the third century, as vulgar Latin gradually became the more common language. (The Wikipedia article on the pre-Tridentine liturgy has some good sources and information, actually.)
In any case, the Kyrie is simply the transliteration in Roman letters of the Greek Κύριε (or Χρίστε), ἐλέησον, “Lord (or Christ), have mercy.” It is all that remains of the original Greek version of the Roman Rite.