It comes from Greek and Latin root-words. In earlier English, it was spelt �Cristes maesse'. The first part of the word is, obviously, 'Christ'...that is, Jesus. This comes from the Greek 'Christos', meaning 'the anointed one'.
The second part 'mas' is an abbreviated form of 'mass'. This word comes from the Latin 'missa', meaning 'sent away', the words said at the end to dismiss the congregation. In this sense, a mass is a religious service, such as those held in the Roman Catholic Church, at which the Lord's Supper is celebrated. It long ago took on the meaning of any church 'service'.
Christmas, therefore - when you put it all together - means "the service held to celebrate the birth of the anointed one, Christ."
As the actual date of His birth is obviously unknown - if He even existed at all! - the Fathers of the Church long ago just decided to tack it on to the festivities that were already celebrated at that time of year to acknowledge the start of days getting longer once more.