I love Christmas carols, especially the old ones. Charles Dickens's story A Christmas Carol is not that old — first published in 1843 — but is written in the form of a "Christmas carol in prose", according to the title page. Its chapters are even called staves. In the first stave, a passing caroler sings a small snippet of an old carol to Scrooge. Do you know the Christmas carol sung in A Christmas Carol?
"God Rest Ye Merry, Gentlemen" goes back to the 1650s, but songs have been associated with mid-Winter holidays for over 2,000 years. For example, the Roman holiday Saturnalia was associated with song, as well as wine and political incorrectness — though it should not be confused with Bacchanalia. There's even a modern Saturnalia song, sung in Latin, titled "Io, Saturnalia" (In English: "Yo, Saturnalia") which might be better to skip.
Carols are not necessarily religious, but they are almost always happy music you can dance to. "O Tannenbaum" means "Oh, fir tree" in German but is usually translated into English as "Oh, Christmas Tree". Other than the word "Christmas", the song has little to do with religion. It just praises the fir tree's "faithfulness" — its ability to stay green all Winter. In German, in French, and in English.