The name Christ translates to Christos in Greek, and the Greek first letter of Christos is "Chi", which translates into the English letter "X." In the case of the term "Xmas," the letter "X" serves as an abbreviation for the word Christ, according to the BBC.
An earlier abbreviation of the word Christmas was XPmas, which appears in ancient documents as far back as A.D. 1021. It is likely that the abbreviations developed as a way to save time and space when creating manuscripts, which were reproduced by hand on costly parchment. It also is possible that religious reasons prohibited scribes from reproducing the full name of Christ.