I was raised Russian Orthodox, but never really understood the technical differences between this type of Christianity and its counterparts, primarily the differences in the calendar that Orthodox Christians use. The origins of Orthodox Christianity would also be interesting.
There's a very simple answer to why most Orthodox churches celebrate Christmas on this date, it's that Orthodox churches calculate such holidays based on the Julian calendar, not having implented the correction of the Gregorian calendar in their liturgical calendar. This results in a 13 day difference between the two.
However, some Eastern groups of Christians (I'm thinking Armenian churches specifically) originally celebrated Christmas on Jan 6th anyway, and so now in fact celebrate on Jan 19th.
The origins of Orthodox christianity is a much more involved answer that I don't have time for right now, but that's the answer to the calendar question.
I can't and won't speak to the origins of Orthodox Christianity, but I can explain why they celebrate Christmas (and everything else) 13 days later than other Christians.
As I've explained previously and as has been covered in AskHistorians before, most of the modern world uses the Gregorian calendar to measure dates and years. The main reason the Catholic Church adopted the Gregorian calendar was to make sure that it celebrated Easter at the right time. The Julian calendar which was being used up to that time was not staying accurately aligned with the seasons: it was losing about 3 days every 400 years. So, between the Council of Nicaea in 325AD where the official timing of Easter was first set, and the time of Pope Gregory XIII around 1580AD, the calendar had lost about 10 days. Where the official date of the vernal equinox was supposed to be 21st March, this equinox was really occurring on 10th March. The calendar was running slow.
So, in 1582, Pope Gregory XIII announced two things:
The Catholic Church would start using this new calendar, effective immediately.
To correct the actual date, 10 days would be "skipped" in October 1582: the 4th October would be followed by the 15th October. This would ensure that the next vernal equinox would fall at the "proper" date of 21st March 1583.
However, even though the Gregorian calendar was first adopted by the Catholic Church in 1582, this had no effect outside the Church: Pope Gregory was merely making a rule for the Church. He had no authority to change governments' civil calendars. However, the Catholic countries of Spain, Portugal, and the Polish–Lithuanian Commonwealth, plus most states in Italy, adopted this new calendar immediately.
And, gradually, over the next few centuries, more and more European countries adopted this new "Gregorian" calendar. However, two of the European countries to hold out the longest were Greece and Russia. These two countries didn't switch over to the Gregorian calendar until the 1920s. By this time, the Julian calendar was 13 days behind the Gregorian calendar. So, when Greece and Russia switched over to the new calendar, they had to drop 13 days to align their dates.
However, even though the governments of Greece and Russia changed the civic calendars, this had no effect on the religious calendar of the Orthodox Church, which has a large membership in these countries (among others). The Orthodox Church decided not to change calendars, and still uses the Julian calendar to measure its religious festivals. And, since dates in the Julian calendar are currently 13 days behind dates in the Gregorian calendar... "their" 25th December falls 13 days later than "our" 25th December.
This difference will continue to increase, by about three-quarters of a day per century. One hundred years from now, the Orthodox Christmas will be 14 days after the western Christmas. Two hundred years from now, it will be 15 days later. And so on.