Usually there is controversy at this time of year regarding "keeping Christ in Christmas", 'Merry Christmas' vs. 'Happy Holidays' as a seasonal greeting and occassionally even calls to boycott businesses that do not use "Christmas" in their holiday advertising. Personally I am not offended by people wishing me a Merry Christmas, although as a pagan I do not observe Christmas or celebrate the birth of Jesus; nor do I try to push my own views upon people. I'm just as likely to use the word 'Christmas' as I am to say 'Yule'. To me, it's not a big deal; to each his/her own. The problem begins for me when certain folks, usually Christians, attempt to bully others into seeing things the way they want to see them, and try to make using the phrase 'Happy Holidays' some kind of evidence of Satan-worship or moral depravity. To a lesser extent, adherents of what is derisively called 'political correctness' work at erasing all vestiges of Christmas and indeed Christinity from our culture, despite it being the faith, however splintered, of the majority of Americans; I'm not a fan of that either.
The truth is that, despite the status of Christianity as the majority religion, there are significant numbers who are not Christian, and their beliefs and customs should be respected as well. Even within Christianity there are those who don't observe Christmas, either because they literally interpret the injunction to "be not observers of days nor seasons" or because they believe that Jesus was born at a time other than Decembert 25th. For a large chunk of my life I believed both: that setting some days aside as special was unbiblical and that Jesus was actually born in the early evening hours of September 11th, 3BC.
So just when did Christmas become Christmas?
There is no universally agreed upon date of Jesus' birth in the bible. Scholars have proposed a variety of dates based on clues in the gospels according to Matthew and Luke, although some hold that the story in Luke, including the visit from the Magi, refers to a time when Jesus was about two years old and is not referring to the events surrounding his birth. Even in the early days of Christianity a variety of days were proposed, eventually settling on December 25th.
Prominence of Christmas celebrations waxed and waned throughout the centuries, increasing in popularity after Charlemagne, and later other monarchs, was crowned on Christmas Day. Eventually Christmas celebrations were marked by partying of all kinds, including drunkeness, gambling, promiscuity and other forms of debauchery...much of what we've come to expect during New Year's Eve parties.
The Protestant Reformation changed things. Puritans saw Christmas celebrations as a "Popish", i.e. Catholic corruption and even banned it at one time, Puritan influence in colonial America resulted in little Christmas observance in the early days of the United States. After independence, Christmas celebrations were discouraged as being "English", although it was celebrated freely in areas of greater German influence.
In the 1820's in England many of the religious tensions of earlier times had somewhat abated. Several writers, including Charles Dickens, pushed for a revival of the imagined "traditional" Christmases of earlier times, seen as a family-centered time of generosity. In England and North America during this time church attendance had fallen and the community and church centered observances had all but disappeared. Dickens' A Christmas Carol, an instant success, helped fuel this new slant on Christmas. In the United States author Washington Irving accomplished much the same purpose with several of his writings where he described Christmas traditions from England which it is suspected that he made up! Clement Clarke Moore wrote A Visit From St. Nicholas, popularly known as 'Twas the Night Before Christmas, cementing the image of St. Nick, aka Santa Claus firmly in the minds of Americans. Many of the modern Christmas "traditions" spring from the minds of these authors.
The time of Christmas itself had long had significance in pagan circles throughout the ancient world. the Romans celebrated the birth of the invincible sun, and many northern European cultures observed the winter solstice (which before calendar reformation was December 25th, not December 21st/22nd) where the lenghthening of days as well as the rebirth of the slain sacrificial god-king was celebrated. Much Christmas imagry, including holly, the Yule log, mistletoe, Christmas trees, ornaments and even Santa Claus himself comes from pagan observances, mostly in northern Europe.
So yeah, Christians celebrate Jesus' birth on December 25th, but the holiday season also includes observances by other religions, including the Jews' Hannukah and pagan solstice; give and receive those "Merry Christmas" greetings, but don't get bent out of shape over "Happy Holidays".