This process is sometimes referred to as "fixing" Rosh Hashanah. However, the Jewish New Year is in Tishri, the seventh month, and that is when the year number is increased.
If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look. This concept of different starting points for a year is not as strange as it might seem at first glance.
In ancient times, the new months used to be determined by observation.
In addition, Yom Kippur should not fall adjacent to Shabbat, because this would cause difficulties in coordinating the fast with Shabbat, and Hoshanah Rabbah should not fall on Saturday because it would interfere with the holiday's observances. (Before the Common Era), which are commonly used by scholars today.
A day is added to the month of Cheshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening. The "first month" of the Jewish calendar is the month of Nissan, in the spring, when Passover occurs.
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.
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.