Depending on the shift of the calendar, the event of the winter solstice occurs between December 20th and 23rd each year in the Northern hemisphere, and between June 20th and 23rd in the Southern hemisphere. It is supposed to be the shortest day or the longest night of the year. Though the Winter Solstice lasts an instant, the term is used to refer to the full 24-hour period.
As Wikipedia puts it - "Christmas is an annual holiday that celebrates the birth of Jesus... The date of the celebration is traditional, and is not considered to be his actual date of birth. Christmas festivities often combine the commemoration of Jesus' birth with various customs, many of which have been influenced by earlier winter festivals."
After the Romans embraced Christianity, the church thereby offered people (many of whom were new converts to Christianity still devoted to their pagan traditions) a Christian alternative to the pagan festivities that re-interpreted many of their old pagan symbols and actions in ways acceptable to Christian faith and practice.
Not just Christmas, but several Christian holidays were put around the pagan dates and customs, and replaced with Christian names by the Church in Rome. For example:
Valentines Day (pagan title, Imbolgc)
Easter (pagan title, Ostara)
Halloween (pagan title, Samhain)
Christmas (pagan title, Yule)
Edward Gibbon writes: "The Roman Christians, ignorant of his (Christ's) birth, fixed the solemn festival to the 25th of December, the Brumalia, or Winter Solstice, when the pagans annually celebrated the birth of Sol"
Grolier's encyclopedia mentions: " .. under the Emperor Aurelian, Rome had celebrated the feast of the 'Invincible Sun' on December 25th. In the Eastern Church, January 6th, also associated with the winter solstice, was initially preferred. In course of time, however, the West added the Eastern date as the feast of the Epiphany, and the East added the Western date of Christmas."