There certainly were some early celebrations. Early writings including a "feast calendar" written in 243AD indicate that there were some celebrations in the third century and perhaps in the second century. Christmas celebrations did not gain widespread prominence however until the Middle Ages or starting from around 400AD.
These celebrations began and occurred in the areas where Christianity started and spread, the areas around the Mediterranean, including the Middle East, North Africa, southern Europe.
Christmas day, December 25, is not necessarily the actual date of Christ's birth. The true date may not be known because we lack enough information to pinpoint it precisely. (Beware that there are many who claim to have calculated the actual date and are most likely no more accurate than December 25.) Other dates including December 25 were used for this feast.
December 25 was settled on by the early church for reasons that are not absolutely clear (but there are some logical reasons for that date). There were pagan celebrations on and around December 25th. There were pagan holidays year round.
There are many theories surrounding Christmas, such as the belief that it was created simply to convert pagans and/or replace the pagan celebrations around the winter solstice. The problem with these theories is two-fold. First, there is a lack of evidence. There was no big push in early church to create a birthday celebration for Christ. Of course, the church was and is "in the business" of spreading Christianity so there is no doubt that the early church fathers wanted pagans to convert and wanted celebrations to honor God.
Second, pagan worship practices were seasonal and on-going. That is, almost any date or time frame could be said to be a pagan "Holy" day. That Christian Holy Days would clash with and even replace pagan days was inevitable.
In short, Christmas began in the early areas of Christendom in the 2nd or 3rd Centuries.
It is also Jesus Christ's Birthday