This is a fair question: what exact event is the date December 25th to commemorate? The answer is not staightforward, and unless one delves into the past - and especially the assorted manipulations of our calendar, one cannot claim to know very much! There are in fact two serious ways in which the deviations of any given prescribed date can occur: i) via a year or years error (which is major) and a date error.
The error of type (i) throws out presumptive times for any given event. For example, if we know that Abe Lincoln was assassinated on April 15, 1865, we would have serious issues if the real date were actually 4 years earlier or later. The reason is that it would throw the timeline for all other events off as well. The error of type (ii) is perhaps not as serious at first glance, unless the date is intended to be specially commemorative of some major event, like a birth or death. In the case of our calendar both types of error have transpired.
1. The Roman Republican Calendar:
The earliest form of this calendar probably had 10 months but a later improved version, but this evolved to a later 12-month form based on the Moon's synodic period (e.g. the time from the same phase to the same phase, say full Moon to next full Moon). Because it was lunar synodic, where the Moon's period is 29.5 days, it meant that 12 months were set out with alternate durations of 29 and 30 days. That meant a total: (6 months x 30 days/month) + (6 months x 29 days per month) = 180 days + 174 days = 354 days.
Simple arithmetic shows that given this - relative to our own calendar (and the assigned length of the year being 365 ¼ days), one would find a whole month's (30 days) difference would accrue after about 3 years (e.g. 3 year x 11 days/yr = 33 days).
A more enhanced version of the Roman Calendar in use from 70 B.C. made small adjustments to the individual months such that the assigned days became (for each month):
For a total of 355 days. So again, as before a whole month deficit accumulated after every 3 years. Since the original version was formed around 2 B.C. this meant a lot of time elapsed before the Julian calendar reform of 46 B.C. (In this reform, driven by the advice of the Alexandrian astronomer Sosigenes, the lunar synodic month was abandoned and the resulting calendar based on the tropical year - determined then to be 365 ¼ days.
For a total of 44 years elapsed this meant a total period displaced:
(44 years x 10 days/ yr.) = 440 days or about 1.3 years on average when other small errors are averaged including odd intercalations of the calendar, i.e. declaring as full years those periods (which may have been only 9 months) when the intercalator's friends were holding public office.
Thus, by the time of the Julius Caesar's redo the calendar was already out of whack by over a year. This meant the putative date for Christ's birth was not 0 A.D., but had to be earlier than 1 B.C. (This, of course, may have been one reason why the antiquated forms of 'B.C.' and 'A.D.' were eventually dropped for C.E. (common era) and B.C.E. (before common era).
2. Alterations in the Julian Calendar:
The main change in devising this calendar, as noted previously, is that a 365 ¼ day year was adopted, which meant tallying 365 days normally, then 366 days every fourth year, which would be a "leap year'.
Another alteration was needed to re-adjust the date of the vernal equinox which had fallen badly out of place under the Roman Republican (lunar) calendar. Of particular import was the vernal equinox or first day of spring, which was originally March 25th, but had devolved to March 22. In addition, since March 25th was the vernal equinox it meant that December 25th was similarly the winter solstice.
This held special import on account of the tie in of the solstice - when days begin to grow longer, with the birth of the Sun, and hence the Roman Sun god. Thus, near midnight of the 24th of December the followers of Mithra would be heading to their special temple on the Vatican (Vatican hill) and declare joy at midnight for the birth of their Sun savior - born in a cave (like so many Sun gods) and to free men from sin, or save them. As noted by author and historian Joseph McCabe (The American Atheist, Nov.-Dec., 2007, p. 9):
"The Savior Mithra had been in possession for ages of December 25th for his birthday. He was the real 'unconquered Sun': a sun god transformed into a spritual god with light as his emblem and purity his supreme command."
In effect, an intercalation by Caesar of the Julian calendar would preserve not only March 25th as the vernal equinox but December 25th as the original birth of Mithra the Sun god. To do this, Caesar intercalated 3 extra months in the year 46 B.C. bringing its length to 445 days. For this reason, 46 B.C. became known as the "year of confusion". (After Caesar's death in 44 B.C. the month Quintilis was named in his honor, hence, July).
3. The Christian Emergence and Further Changes
The Edict of Milan, formalized in 313 A.D. returned confiscated property to Christians (hitherto under persecution) and more importantly, allowed religious toleration so that Christians could gather without fear of interference or challenge. This directly set the stage, along with other steps, to unite the Christian church with Sun god religion of Constantine. Calendrical alterations directly preceded this 'takeover' of one god by another.
The first alteration in the wake of the Edict was the Council of Nicaea which, in 325 A.D. defined the dates of Easter and certain other religious holidays. In particular, March 21 was re-specified as the date of the vernal equinox while Easter was defined to occur on the first Sunday after the 14 day of the Moon (e.g. days after the full Moon). It ought to be noted here that the Christians at Nicaea didn't willy- nilly just change the date of the vernal equinox from March 25 back to March 21. No, what happened is that between 45 B.C. and 325 A.D. that date had slipped back from March 25th to March 21st. The reason is based on simple math: Because the Julian year (Sec. 2) was defined with an average of 365 ¼ days and is 11 mins. 14 secs longer than the tropical year of 365 days 5 h 48 min 46 ec then the slight difference had accumulated to 3 days in those 4 centuries.
Of course, in the same interval, the original winter solstice of December 25th had also regressed to the earlier date of Dec. 21-22 (depending on the year).
It wasn't until 29 years after the Council of Nicaea, in 354 A.D. that the newly liberated Christians claimed December 25th - the designated original birth date of the Sun god Mithra- as their own nativity for Christ. There are many speculations as to why Christians appropriated this particular date when the eastern orthodox and other variants stuck with January 6th. The most plausible one is probably because the conflation allowed Christian Rome to pay homage to the earlier Emperor Constantine who had expedited their liberation. By allowing the key dates for the Sol Invictus cult and their own deity to coincide, the emerging Christian church not only gained further credibility for their own god (while pacifying many skeptical Romans) but also timed their celebrations to coincide with the Roman "Saturnalia" - which emphasized good will to all men and "peace on Earth" (so no wars could commence)
4. The Gregorian Calendar.
By 1582, that tiny deviation between Julian and Tropical year of 11 mins and 14 seconds had grown to another 10 days so that the first day of spring was occurring on March 11. If this regressive trend were allowed to continue, Easter would eventually fall in the middle of winter. (Bear in mind again, Nicaea assigned Easter not by date of year but by days after the first full Moon after the vernal equinox. If the latter date was earlier, then Easter had to be as well because its timing was bound to the equinox date).
The correction or Julian calendar reform was instituted by Pope Gregory and became known as the "Gregorian Calendar".
The reform here proceeded in two steps:
1) Ten days were dropped off the existing calendar to bring it back to March 21 - where it was at the time of Nicaea. This was undertaken on October 4, 1582 when the next day was proclaimed as October 15, 1582. (Some idiots at the time actually complained that the pope had "taken 10 days out of our lives".)
2) The rule for the leap year was changed so that the average length of the year would closely aprpoximate the tropical year. The rule then applied was that only century years divisible by 400 would be leap years. Thus, 1700, 1800 and 1900 - all leap years under the Julian calendar were not under the Gregorian, while 1600 and 2000 were.
Meanwhile, the average length of the Gregorian calendar, at 365.2425 days, was correct to within 1 day every 3300 years.
Now, WHO says math isn't important?
The upshot of all this is very strange but needs to be stated.
1) Up until 354 A.D. Christians had no remote idea of when their Savior was born. As noted by McCabe (ibid.):
"Early Christendom found itself in the peculiar position of telling the world of the most tremendous birth there ever was...but being unable to say when it happened."
2) The current assigned month, date and year cannot possibly be correct, but must be a confection. As we saw, the designated year is off, and in addition, December 25th marked the original winter solstice and birth of the Roman Sun god, Mithra.
So tomorrow, when people are saying they're celebrating the birth of a "Savior" are they truly aware of which one that is?
Inquiring minds want to know!