During the Late Republic, social disorder, political strife and an ongoing series of civil wars left the maintenance of the calendar in complete disarray. By the time Julius Caesar consolidated his power in 46 BC, the calendar months were off by as much as several months in comparison to the seasons.
Fortunately, while Caesar was in Alexandria, Egypt, he had access to the ancient world's foremost astronomy experts. The Egyptians were able to calculate the length of the actual solar year as 365.25 days and the winter solstice as December 24.
While the Eqyptians did the work, Caesar was the one who both authorized and implemented its use, thereby receiving the credit. Regardless, its accuracy at the time it was developed was remarkable, and only a millennium of a slight annual difference would eventually accumulate. The new Julian Calendar would follow the solar year, rather than lunar, with a total of 365 days vs. 355. The intercalary month was eliminated and the leap year, adding a day to February every 4th year, was adopted as well.
In 46 BC, the calendar had to be set right before the new one could begin. A total of 3 intercalary months were inserted prior to the start of the new year. 46 BC, therefore, or 707 AUC to the Romans, was fifteen months and about 445 days long according to the calendar.
Caesar, in his divine ego, also renamed Quintilis, the fifth month, Julius (July), after himself. His heir and eventual first Emperor, Augustus, did the same to Sextilis, changing it to the name we now know as August. This basic transformation has essentially remained intact for 2 millennia and represents the foundation of the western calendar still in use today.
It's also the common opinion that Augustus moved a day from February to August in order to make his month the same number of days as Caesar's (July). However, there is little actual evidence to support this. We know that Caesar made every even month 30 days (except February) and this would seem to fit for August as the 8 month. No ancient sources seem to mention the movement of the day, even those who came after Augustus, who could've done so without fear of reprisal. While it seems to fit that Augustus added a day to the 30 day even month of August, there just isn't enough support this without a doubt. Even the historian Suetonius, who assuredly printed any rumor or gossip about his subjects without concern over factual data, fails to mention the movement of the day.
"Inasmuch as the calendar, which had been set in order by the Deified Julius, had later been confused and disordered through negligence, he restored it to its former system [8 B.C.]; and in making this arrangement he called the month Sextilis by his own surname, rather than his birthmonth September, because in the former he had won his first consulship and his most brilliant victories."
Unfortunately, as suggested, a slight miscalculation on the part of the ancients left Renaissance scholars in a bit of difficulty. The actual year is 11 minutes and 14 seconds longer than what was calculated by Caesar and his Egyptian astronomers, so over the centuries, the difference grew to as much as 10 days. By 1582 AD, while the difference was still fairly negligible in comparison to seasonal circumstances, Pope Gregor XIII devised a new calendar, simply adjusting the old. The removal of three leap years each four centuries, basically, every year divisable by 100, but not 400, was removed from the calendar going forward.
The difference in future years was then taken care of, but Gregor also had to remove 10 days from the month of October, 1582 to bring things back in line. This reduced the error to almost nothing. The Gregorian Calendar, while more accurate then the Julian, took some time to catch on. Many European nations adopted it rather quickly in the 17th century, but Britain and her colonies, including the United States, waited until much later in the 18th century. Russia was the last of the western nations by doing so in the 20th century.
Overview of the Julian Calender