We don't know when the Jewish religious leaders stopped making yearly decisions about when to intercalate (add extra days to the calendar to adjust for the differences between solar and lunar time), but by the 11th century A.D. it was believed that a standardized calendar had been introduced by Hillel II in A.D. 358-9. An extra, intercalated
month now occurs seven times in each nineteen-year cycle: on the third, sixth, eighth, eleventh, fourteenth, seventeenth, and nineteenth years. The name of the intercalated or "leap" month (Adar II or Adar Bet) is a repeat of the name for the twelfth month (Adar I or Adar Alef), since the intercalated month always falls immediately after the first or "alef" month, alef being the first letter of the Hebrew alphabet. Bet is the second.
Secular vs. Religious New Year
The Jewish calendar is designed around its major festivals, Passover and Rosh Hashanah Rosh (ראש השנה)
. Rosh Hashanah, 10 days before the Day of Atonement, Yom Kippur (together: "the High Holy Days"), is the beginning of the Jewish secular year and comes in the fall, while Passover is the beginning of the religious year and comes in the Spring. While it may seem confusing to have two beginnings of the year, it is no different from what happens today, in the U.S., where the school year begins in the ninth month, September. There are even special calendars for U.S. students based on this school year.
Nisan, which begins at the new moon, is the beginning of the religious year, according to the Biblical Exodus. It is the month from which the Biblical kings dated their reign, and the month in which Passover, the occasion that started the Exodus of the Jews from Egypt, is celebrated on the fourteenth (at the full moon).
The secular calendar begins in the month of Tishri, the seventh month of the religious year. The month of Tishri, like Nisan, always begins at the new moon.
Rosh Hashanah was the first day of the first month of the secular year (Tishri), from at least the time when new moons were proclaimed by the Sanhedrin and its successor, the Jewish Assembly. As Jewish population grew, it spread out. Signal fires had to be relayed. Soon not everyone was aware of the new month on the same day, so Rosh Hashanah came to be celebrated for two days. Today, even with instant satellite communications, Rosh Hashanah continues to be celebrated over two days, possibly because the first glimmer of the new moon is visible in Israel before it is visible in many other parts of the globe.
See: Months of the Jewish Calendar