You'd be surprised what they were singing at Christmas in the Middle Ages.
For other carols, the tunes alone survive and the original lyrics (if there were any) remain unknown. "Good King Wenceslas" is based on a 10th century legend about Bohemian martyr-king Wenceslas I, who once took in a poor woodcutter "on the Feast of Stephen" (December 26). The lyrics were written by John Mason Neale (1818-1866) c.1850. The Swedish tune, however, probably dates to the 13th century and was first published in 1582. It is a spring carol called "Tempus Adest Floridum". The tune was also used for a carol called "Gently Mary Laid Her Child", written in 1919 by Joseph S. Cook.
"The Holly and the Ivy" is a tune of unknown origin with decidedly pagan overtones. The Holly and the Ivy appear to represent men and women respectively, but also may hark back to druidic traditions. The carol appears to be traditionally from Gloucestershire in England, where Cecil Sharp recorded it during the 19th century.
With still other carols, the lyrics survive, but not the tune. Franz Schubert composed the famous tune for "Ave Maria" (Hail Mary), for example, to lyrics by Walter Scott. But since then, the now more famous Latin prayer has been used instead. While this prayer only dates to the 15th century, the Marian cult it represents is, of course, much older, dating nearly to the beginnings of Christianity.
The Latin words of O, Come, O Come Emmanuel date to a 13th century French manuscript; the tune comes from a 15th century processional. Some have tried to date the lyrics to the early 9th century. The song glorifies the birth of Christ (by his Old Testament name "Emmanuel") as a triumph over death and Satan.
Probably the oldest Christmas carol in existence is the "Magnificat" (c.50-100), the song that Mary sings glorifying God after discovering that she is pregnant with Christ in Luke 1:46-55. This is the Annunciation, in which an angel visits Mary and tells her that she is pregnant with the Son of God. Mary then visits her elderly cousin Elizabeth, who is pregnant with John the Baptist. After Elizabeth's child leaps in the womb in response to Mary's approach, one of the two women begins to praise God. This is the Magnificat. It has, naturally, been set to music many times.
See part 1 for more medieval Christmas carols and their origins and go here for reviews of collections of medieval Christmas music.