I'm not sure how, but until this year, I managed to get through over four decades of life without learning that there was a controversy over the use of the word "Xmas" as an abbreviation for Christmas. As is no surprise, this controversy is the creation of evangelical pastors who somehow got through seminary without even a smattering of the history of their faiths. It seems that these angry, but well-meaning, fellows somehow think that the abbreviation "Xmas" was somehow popularized by the heathen to take the "Christ" out of "Christmas" as part of a plot to secularize the nation. There are a whole slew of problems associated with how both Christians and non-Christians perceive this particular holiday, but today we'll stick with this Xmas thing.
The root of the problem is that the spelling of "Christ" is different in Latin-derived alphabets like English than is it is in Greek, the language of the New Testament. In Greek, it is spelled "Χριστός", or "Christos". The first letter is "Chi", or in Greek "Χ". In the ancient and medieval world, this was as obvious to educated folks as it was to Christians. Ancient Christians used the "X" (Chi) and the "P" (Rho) letters as a symbolic shorthand for Christ (the labarum) that still exists within the Catholic Church in its use on vestments and altar service. The Chi-Rho symbol is also that which reportedly appeared to the Emperor Constantine in a dream before the Battle of Milvian Bridge in 312 A.D., where it was later claimed that he had his soldiers paint it upon their shields.
So, at least in the ancient world, the use of the "X" was a common abbreviation for Christ and Christians. The use of the Chi as an abbreviation after that point is a bit more complicated, but historians generally agree that by the fifteenth century, the term Xmas was widely used. Religious abbreviations were common due to the difficulty in reproducing manuscripts in scriptoria, ink was expensive, as was parchment and paper. That meant that in Gregorian chants and other documents, the monks that made copies relied on abbreviations to reduce the time and cost to reproduce their works. Other examples include "Dne" for "domine" and "ala" for "alleluia". We can definitively date various uses of the "X" in abbreviations for Christmas in England (as "Xp̄es mæsse") at least as far back as 1021 when it appears in the Anglo-Saxon Chronicle. The expense of printing the entire word "Christmas" was such that even after the Gutenberg's development of the movable-type printing press in 1453, even the church was using Christmas in its documents.
Of course, even the word "Christos" is a title, not the actual name for Jesus, in either historical or Biblical terms. "Christos" is Greek for "the anointed one". I would be willing to wager that most Americans, even devout Christians, don't understand that distinction when they are arguing about keeping the "Christ" in "Christmas". Of course, there's really a whole different argument about how, why, and when Christmas should be celebrated.