What I have always thought / known is that:
Days are based on the period of rotation of the earth.
Months are based on the moon.
Year is the time taken by the earth to make one revolution around the sun.
But if we take a look at the lunation it takes on average 29 days, 12 hours, 44 minutes, and 28 seconds. Why is it that the months then in the Gregorian calendar have 28-29 / 30 / 31 days?
Ancient Egyptians and Mesopotamians came up with “administrative” calendars of 30 days, that were easier to calculate than “real” lunar months of sometimes 29, sometimes 30 days. At the end of the 12 months, they added five days of religious observances, sometimes positive, sometimes negative.
When Julius Caesar conquered Egypt, he knew about the very regular Egyptian calendar and appointed his mathematicians and astronomers (if we can really call them that, as Roman science wasn’t very developed) to discuss with their Egyptian peers and devise a better calendar for the Roman Republic (reminder that Caesar was not the Emperor; the first one was Augustus, his [adoptive] son, and not right away).
Eventually, it was decided in Rome that, in order to follow with the seasons, the already-existing Republican Roman calendar was to be modified. It contained 10 months of 29 and 30 days plus a “winter” of 59 days, which became divided in January and February. As Julius Caesar was the one who brought this change, he was “honored” by having a month renamed from Quintilis (“Fifth”) to Julius (our July)—and that month was given 31 days, as well as a few others. Also, the beginning of the year was shifted (at this point or later; it’s not clear) to January, making the “numbered months” off with their new rank—e.g. “October” is not eighth (“octo”) anymore.
Caesar’s reform was not well applied, and eventually, Augustus had to enforce the new calendar, and he was also “honored” with a 31-day month, right after Julius’, so that’s how Sextilis (“Sixth”; and it had 30 days) became Augustus (with 31 days).
The rest is basically adjusting here and there to make it a total of 365 days at the end of the year, someone sometime having the “brilliant” idea of alternating 30- and 31-day months (a little like nature “alternates” between 29- and 30-day lunar months).
Side note… In some languages, the word for Moon and the word for month is the same—it’s already quite similar in English! For example, Romanian has “Luna” for both, and Korean has ”dal.”
Calendars can either be lunar or solar.
A lunar calendar has months that match the phases of the moon but years that don't match the Earth's rotation around the Sun. It can't have both because the moon doesn't pass through an integer number of whole cycles in the year. The most common example of this kind of calendar is the Islamic Calendar in which dates move through the year as the years pass. On a given day in the Islamic calendar the moon will always be in the same phase, but the Earth will be in a different position in its orbit around the Sun.
The Gregorian calendar in contrast is a solar calendar. It's years match (at least very closely) to the period of the Earth's rotation around the sun, and a given day - say 6th January - occurs when the Earth is in the same position in its orbit around the Sun, but the phase of the Moon will vary from year to year.
There are also hybrid calendars which try to do both by, for example, adding an additional month every 2.71 years (like the Hindu calendar) to keep the days in a lunar calendar roughly, although not exactly, in sync with the rotation of the Earth around the Sun.
Why exactly different cultures chose one or another is likely as much to do with historical accident as anything else, but there's some suggestion that cultures further from the equator tend more towards solar calendars. But that doesn't have much to do with Astronomy, the core Astronomical reason is simply that the Moon orbits the Earth in a number of days that doesn't exactly divide the number of days it takes the Earth to orbit the Sun.
What made calendar systems (any calendar, any civilization) "messy" is the belief that there is a magic cycle so that all repeating events observed can be described in integer number of this magic "base" period (month, day, hour,...), or at best in fractional numbers (and why in the decimal system by the way? After all, the Babylonians used the sexagesimal system, that we still use for dividing time into smaller units). Another wrong belief is that observed cycles do do not drift over years (everything must come back to the same "point", in a fixed time). Also, a wrong starting assumption is that the observer of the repeating events is in a fix reference, for example a "static" Earth. Even until today, not many people understand the difference between a solar day (86400 seconds) and a sidereal day (86164.0905 seconds). When we understand these wrong beliefs and wrong starting assumptions, we can understand why ancient and less ancient priests (and astronomers) were desperate in their quest of a "nice" celestial order. So, when you try to fit the Earth cycle to the Moon cycle, you will have problem in finding a "nice" mathematical solution.
