Until 45 BCE the Roman calendar consisted of twelve lunar months of 29 or 30 days, each starting on a new moon, for a total of 355 days in a year. This falls a bit more than ten days short of the actual solar year of 365.2422 days, so every once in a while (at the discretion of the pontifex maximus) a thirteenth month was inserted, called intercalans.
The maintenance of the calendar and its correspondence with the lunar and solar cycles had been neglected for a long period during the civil wars of the late republic, so the discrepancy had accumulated to several months. Meanwhile, Caesar‘s stay in Alexandria in Egypt had put him in contact with expert astronomers. They were able to calculate the length of the solar year to 365.25 days and the winter solstice to the 24th of December.
The new calendar would follow the solar year and have twelve months of 29, 30 or 31 days, for a total of 365 days, with no intercalary month. Every fourth year the month of February would be one day longer, so an average year would be 365.25 days. The new calendar should have started on the day of winter solstice, but the new moon was seven days later, and ancient tradition said the kalends should be on a new moon, so the first of January was postponed seven days.
The year 46 BCE was a transitional year. The discrepancy from the intercalary months that had been ignored had accumulated to 90 days, so after November of 46 BCE first a normal intercalans was inserted, followed by two more intercalary months to make up for the difference. The result was a year, the last year of confusion, of fifteen months and 445 days!
With this reform the fifth month of the year (counting from March), Quintillis, was renamed after Julius Caesar: July. Augustus later renamed Sextillis, the sixth month, after himself, and moved one day from February to August so his month would be as long as Caesar’s, thus arriving at the basic calendar we still use today.
There is a slight defect in the Julian calendar, since the year is 11 minutes and 14 seconds too long. This discrepancy too accumulated over the centuries, and by 1582 CE it had become 10 days. Pope Gregor XIII decreed the removal or three leap years each four centuries, so years divisible by 100 are not leap years unless they are also divisible by 400. This reduced the error to almost nothing. Gregor XIII then removed 10 days from the month of October, 1582 to bring the year back in line with the solar cycle. The adoption of the Gregorian Calendar happened at various times throughout Europe. Most countries had adopted it by the 17th century, England and its colonies in the 18th century and Russia only in the 20th century.