January is the first month on the Gregorian and the Julian calendar. It is named after the Roman god of doors, Janus, as this month is a doorway into the new year. Janus is an interesting Roman god as he is two-faced. Thus, he can see both the future and the past. In January, you can see the previous year and view the upcoming one. Prior to the Julian calendar, the calendar was set by lunar rather than solar days. This resulted in problems creeping in and causing confusion. Also, the start of the new year was in March since spring started in that month. This meant that January and February were the last two months of the year on the old Roman calendar.

The problem with the old Roman calendar
Since the calendar relied on lunar rather than solar days, it was three months ahead of the solar based calendar. Which meant if you used one calendar for civic events but the other for your growing season, then obviously problems would arise. Caesar was advised by Sosigenes, an Alexandrian astronomer who helped create the Egyptian solar calendar, to also make the Roman one based on the sun rather than moon. Under his plan, the year was divided into 12 months and each month had either 30 or 31 days. He calculated a solar year as 365 ¼ days. February was the exception by having 28 days and every four years having a leap year to add an extra day. Due to misunderstandings and other issues, it would not come into effect until 8 BC.

Sosigenes calculations turned out later to be off by 11 minutes and 14 seconds. That would seem trivial but, over time, the cumulative effect was a 10 day difference from Caesar’s time. Which meant the calendar was no longer aligned with the solar year and had to be rectified. This caused problems with celebrating holy days that needed to be calculated precisely according to astronomical and other calculations. The drift was noticed in the Middle Ages and calls to correct were made. At the Council of Trent (1545), Pope Paul III was authorized to reform to calendar to allow for a more consistent scheduling of Easter.

While several reforms were suggested, a proposal made by Aloysius Lilius offered a reform that was considered acceptable. His proposal was to reduce the number of leap years in the past (making them common years rather than leap). And then he had an idea of adjusting the phases of the moon (meaning a method to calculate the difference between solar and lunar years) when calculating the annual date of Easter. This had always been a problem in the past and his solution seemed to resolve it.

Gregorian Calendar Introduced
In October 1582, Pope Gregory XIII issued the reform of the Julian calendar. It was adopted by the Catholic Church and the Papal States. Since however this was a civic reform, it was up to each nation to decide whether to implement or not. It would gradually be adopted by many countries. Spain was the first to switch over and that included much of Roman Catholic Europe. Protestant countries were not keen on changing right away since the reform was made by the Catholic Church. The British would adopt it 1750 but by a method to avoid saying it was from the Catholic Church. Sweden adopted in 1753. Turkey would switch to using the fiscal year as Gregorian in 1917 and then for the entire calendar in 1926. Russia, under the Communist government, changed in 1918. Greece would change in 1923. Saudi Arabia would formally adopt it in 2016.

Eastern Orthodox denominations decided for religious purposes to use the Julian rather than Gregorian for their liturgical year (separate from the civic calendar). Which is why in countries like Greece or Russia the celebration of Christmas and Easter is currently 13 days after it is celebrated elsewhere.

Sources:

Britannica.com
Catholic Encyclopedia
Timeanddate.com