A Christmas carol is a carol

Carol (music)

A carol is a festive song, generally religious but not necessarily connected with church worship, and often with a dance-like or popular character....

  (song

Song

In music, a song is a composition for voice or voices, performed by singing.A song may be accompanied by musical instruments, or it may be unaccompanied, as in the case of a cappella songs...

 or hymn

Hymn

A hymn is a type of song, usually religious, specifically written for the purpose of praise, adoration or prayer, and typically addressed to a deity or deities, or to a prominent figure or personification...

) whose lyrics are on the theme of Christmas

Christmas

Christmas or Christmas Day is an annual holiday generally celebrated on December 25 by billions of people around the world. It is a Christian feast that commemorates the birth of Jesus Christ, liturgically closing the Advent season and initiating the season of Christmastide, which lasts twelve days...

 or the winter season in general and which are traditionally sung in the period before Christmas.
The first specifically Christmas hymns that we know of appear in fourth century Rome

Rome

Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...

. Latin hymns such as Veni redemptor gentium, written by Ambrose

Ambrose

Aurelius Ambrosius, better known in English as Saint Ambrose , was a bishop of Milan who became one of the most influential ecclesiastical figures of the 4th century. He was one of the four original doctors of the Church.-Political career:Ambrose was born into a Roman Christian family between about...

, Archbishop of Milan

Milan

Milan is the second-largest city in Italy and the capital city of the region of Lombardy and of the province of Milan. The city proper has a population of about 1.3 million, while its urban area, roughly coinciding with its administrative province and the bordering Province of Monza and Brianza ,...

, were austere statements of the theological doctrine of the Incarnation in opposition to Arianism

Arianism

Arianism is the theological teaching attributed to Arius , a Christian presbyter from Alexandria, Egypt, concerning the relationship of the entities of the Trinity and the precise nature of the Son of God as being a subordinate entity to God the Father...

.