We have today three versions of the calendar measuring our days and weeks and years.
The first is the Biblical calendar instituted by God:
"Now the LORD said to Moses and Aaron in the land of Egypt, "This month shall be the beginning of months for you; it is to be the first month of the year to you. Speak to all the congregation of Israel, saying, 'On the tenth of this month they are each one to take a lamb for themselves." (Exodus 12:1-3 NASB unless noted)
The calendar year was to begin in the month of the passover, in fact, ten days before the passover.
By God's directive, this calendar used numbers to designate the months, i.e. 1st month, 2nd month, 3rd month, etc.
God's 1st month corresponds with around the last week of March of the Gregorian calendar and the month of Nissan on the Jewish-Babylonian calendar.
The second Calendar to come to the fore after the exile in Babylon. This Calendar had as it’s first month the month of Tishrei, about 5 months after the Biblical Calendar’s first month. The names of the months are Tishrei, Cheshvan, Kislev, etc., actually Babylonian names for the months.
Both the Biblical and Babylonian calendars are based upon the Lunar cycle. Prior to the events on Hezekiah’s days (2 Kings 20) the Earth had a 360 day cycle. This gave 12 lunar months exactly. At that time the earth was moved to the present 365.25 days. This made it necessary to add an extra month 7 times in a 19 year cycle. This is the reason and meaning behind the 12/13 month leap month.
In 1582 the world was introduced to the Gregorian Calendar which most western nations are familiar with today. This replaced the Julian Calendar which was less accurate. The Gregorian Calendar consists of 12 months of 30 or 31 days with the exception if February which was 28 days. Every 4th year was a leap year wherein an extra day is added to the month of February.
The names of the months vary in meaning with some named after Greek/Roman gods (Mars), Roman rulers (August - Caesar Augustus), and mis-labeled Latin names or numbers (October, November, and December in Latin mean 8th, 9th, and 10th months). These numbered months were bumped up two places to make room for the Caesars, Julius and Augustus (July and August).