You have known this melody since childhood; now meet it in its own language. Verse one and the tune are usually credited to John Francis Wade, an English Catholic music copyist who wrote them out in the 1740s and first printed them in 1751; the shepherds' verse we print second was added in 1794 by the French priest Étienne Borderies. The carol has never been out of use since.
The refrain is a grammar drill in disguise: adoremus - come, let us worship - pairs a plural imperative with the hortatory subjunctive, and the two verses on this page make you sing adoremus six times (seven, counting festinemus). By the last refrain, -emus meaning "let us" will be wired in. The second verse opens with the cleanest ablative absolute in the songbook: grege - the flock having been left behind - two words carrying a whole clause, which is exactly the trick that makes Latin Latin.
The rendition we link is Pavarotti, whose Latin diction is superb and firmly ecclesiastical: Regem as REH-jem where the classical course audio would say REH-gem. Enjoy the difference - it is a thousand years of the language's afterlife in one consonant.