Bible History Daily has a nice essay, "How December 25 Became Christmas."
A few points are:
- "Celebrations of Jesus’ Nativity are not mentioned in the Gospels or Acts; the date is not given, not even the time of year."
- "The extrabiblical evidence from the first and second century is equally spare: There is no mention of birth celebrations in the writings of early Christian writers such as Irenaeus (c. 130–200) or Tertullian (c. 160–225)... As far as we can tell, Christmas was not celebrated at all at this point."
- Easter, on the other hand, was celebrated. "Jesus’ ministry, miracles, Passion and Resurrection were often of most interest to first- and early-second-century C.E. Christian writers. But over time, Jesus’ origins would become of increasing concern."
- "Finally, in about 200 C.E., a Christian teacher in Egypt makes reference to the date Jesus was born."
- "By the fourth century, however, we find references to two dates that were widely recognized—and now also celebrated—as Jesus’ birthday: December 25 in the western Roman Empire and January 6 in the East (especially in Egypt and Asia Minor)."
- "So, almost 300 years after Jesus was born, we finally find people observing his birth in mid-winter. But how had they settled on the dates December 25 and January 6?"
- "The most loudly touted theory about the origins of the Christmas date(s) is that it was borrowed from pagan celebrations." This is false. See the article for a full explanation.
- For how December 25 was settled on see the article.