what is the symbol of carols?

A carol was originally any song associated with any feast day which was also intended to be danced to. Think of it as the medieval version of club music.

"History tells us that carols have no connection with Christmas or even with Christianity for that matter. The melodies were originally written to accompany an ancient dance form known, as the 'Circle Dance'. The dance form was mainly associated with fertility rites and pagan festivities in the medieval Celtic countries of Europe. The Christian Churches, which were established in these areas, got familiar with the melodies and rhythms of carols and they easily found their way into Christian meetings and celebrations. The songs had such polytheistic and pagan roots, that the Churches were not comfortable about them for a long time. In fact, in the mid-Seventh Century the Church Council explicitly forbade Christians to sing carols, and it continued till the Twelfth Century but with the passage of time a kind of renaissance took palace and the carols were fused with the folk songs that were the Pop songs of the day - the songs that were whistled or sung by ordinary people. It is said that Saint Francis of Assisi is the person who brought the change. The priests in St. Francis' developed a different style of religious folk song called a lauda. The tunes of the songs were so catchy that it soon spread across the Fourteenth Century Europe. The music evolved with time and became so popular that even today carols are so much a part of Christmas celebration some seven centuries later." http://christmas.indiaserver.com/christmas-carol/i...

This, along with most "Christian" celebrations are based on other religions' traditions that were basically taken over by Christianity to make conversion easier. Christmas itself is not Christian, as many Biblical scholars have found that Jesus' supposed birthday was in a different season altogether. In ancient Rome, a celebration in honor of Mithras, a Persian sun god, was celebrated right after the winter solstice. The early Christians did not want to celebrate such a "pagan" holiday and the leaders in Rome made a compromise between the two factions creating a holiday on December 25. As Rome conquered much of Europe, the celebration made its way into other religion's celebrations (which is why we have large feasts, decorations of greenery, and gift-giving during this time of year).