The Hebrew calendar looks much like the Babylonian one, and there is clear influence from the time of the Babylonian captivity. In its earliest stage, the months were numbered, rather than named. The names eventually adopted were versions of the Babylonian names. As in Babylonian reckoning, Nisan was originally the first month of the year. Tishri became the first month along with the western part of the Selucid empire, and it remains so today.
Despite this obvious Babylonian influence, Jews did not adopt the regular 19-year cycle for inserting intercalary months, nor did they use Babylonian mathematical calculation of new moon. The decision to insert an extra month was made by the Sanhedrin in Jerusalem on rather vague criteria such as the appearance of new plants. Because they measured neither the equinox nor helical risings, the old Hebrew calendar cannot be reconstructed by mathematical formula.
Another difference between the Hebrew and the Babylonian calendar is the treatment of the 7-day cycle. Recall that the Babylonians had a 7-day cycle, but the days around the new moon when it was invisible were not included. In the Jewish scheme, the 7-day interval between Sabbaths runs independently of the months and years. There are no epagomenal days. The days are numbered 1 to 7. Only the Sabbath, the seventh day, is named, although day 6 is sometimes called ereb shabbat, "the day preceding the Sabbath."
In the 2nd and 3rd centuries CE, the Jewish calendar was reformed. The primary purpose of this reform was to regularize the intercalation of months and the length of the months. Using the Metonic cycle of 19 solar years, months are intercalated in years 3, 6, 8, 11, 14, 17, and 19 of the cycle, exactly the same spacing as in the Babylonian cycle. In a regular year, the months alternate between 30 and 29 days (Tishri has 30, Marheshvan 29, etc.). The embolismic month has 30 days, and intercalated between Adar and Nissan (never after Elul). It is called Second Adar, or Weadar. Nissan still has 30 days.
Certain customs about the days of the week upon which the High Holy Days may be celebrated require adding a day to certain years and then subtracting a day from the following year. The shorter years are called "defective", and Kislev is decreased to 29 days. The longer years are called "perfect," and increase Marshevan to 30 days. Those interested in the precise formulation of these rules should consult the references.
The epoch that Hebrew calendar currently uses, the Hillel world era, begins October 7, 3761 BCE. This epoch was calculated by Hillel II in the 4th century CE, but did not become universal practice until the end of the Middle Ages. Other epochs used before then were the so-called era of Adam (3760 BCE), and the Selucid (312 BCE).