“The calendar”, in the sense of the solar calendar of 12 months and 365 days starting on January 1 that much of the world uses today, was established by Julius Caesar in 46 BC.
The calendar was probably originally a lunar calendar, far in the pre-historical Roman past. At some point the months became standardized in length, with 29 or 31 days each. This version of the calendar was attributed to the legendary king Numa Pompilius, the successor of Romulus, who was supposed to have ruled in the 7th century BC.
The year always seems to have started in January, followed by the short month of February, then March, April, May, and June. They were followed by months that look like they were numbered from 5-10 (Quintilis, Sextilis, September, October, November, December), even though they were really 7-12. The most logical explanation is that Quintilis-December were originally the 5th to 10th months, then January, then the short month of February at the end. But aside from logic and the names of the months there isn’t really any documentary evidence for a calendar with the months in that order.
The Romans attributed the placement of January at the beginning to the Second Decemvirate, a council of ten men who also published the Twelve Tables, Rome’s earliest legal code, in the 5th century BC. The change were also attributed to Gnaeus Flavius, an aedile in the late 4th century BC, who published the calendar on tablets in the Forum in the 4th century BC. But these were just guesses - the calendar started in January, but no one knew why, that’s just the way it was.
Regardless of whether the year started in January or March, the problem was that Numa’s calendar only had 355 days, even though it was obvious that the solar year actually had 365 days. At first, the priests in charge of maintaining the calendar added an extra month of 22 or 23 days every two years. But for whatever reason, sometimes…they just didn’t do it. There might have been religious or political reasons not to do it - maybe one year the priests decided it would be inauspicious to add more days, or the political situation disrupted their activities, or the political leaders of the city decided not to add more days. Over the centuries, disruptions like this caused the calendar to drift from the solar year. The 1st century BC was especially chaotic, so by 46 BC, the Roman year had drifted by a few months.
Caesar reorganized the calendar with the now-familiar 12 months of 30 or 31 days (or 28) for a total of 365 days. To make the year line up with the solar year, three intercalary months had to be added for the year 46 BC, which ended up with 15 months and 445 days. By now the Romans also realized that the solar year isn’t exactly 365 days, it’s actually about a quarter of a day longer, so Caesar added extra day at the end of February every 4 years (starting in 44 BC). Quintilis was eventually renamed Julius/July after Caesar, but not until after he was assassinated (also in 44 BC). His adopted son, the first emperor Augustus, thought he deserved his own month too, so he renamed Sextilis.
This calendar was known as the Julian calendar and it was so successful that it was used until the 16th century. By then it was known that simply adding a leap day every 4 years wasn’t exactly correct since the solar year is just slightly longer than 365.25 days. The 16th century calendar had drifted again, by 10 days. The reorganized calendar is now known as the Gregorian calendar, after the pope at the time, Gregory XVI.
So, the short answer is, the Roman calendar was published and maintained each year by priests who didn't always properly carry out their calculations. By 46 BC, Caesar determined that the year needed to be 445 days long in order to bring it back into harmony with the solar year.
Sources:
Jorg Rupke, The Roman Calendar from Numa to Constantinople (Wiley-Blackwell, 2011)
Alan E. Samuel, Greek and Roman Chronology: Calendars and Years in Classical Antiquity (Munich, 1972)
Denis Feeny, Caesar’s Calendar: Ancient Time and the Beginnings of History (University of California Press, 2007)