Sun dating phenomena
Years are either 12 or 13 months, corresponding to the 12.4 month solar cycle.
The lunar month on the Jewish calendar begins when the first sliver of moon becomes visible after the dark of the moon.
The Jewish calendar is based on three astronomical phenomena: the rotation of the Earth about its axis (a day); the revolution of the moon about the Earth (a month); and the revolution of the Earth about the sun (a year).
Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.
This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.
Note that most of the Bible refers to months by number, not by name.
The Jewish calendar has the following months: The length of Cheshvan and Kislev are determined by complex calculations involving the time of day of the full moon of the following year's Tishri and the day of the week that Tishri would occur in the following year.