(from a manual of Jewish customs,
Amsterdam 1695)
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The Bible speaks often of days, months and years, but does not define them.
[Note that the "week" does not reflect a natural or astronomical cycle, and is only found in Judaism or in cultures influenced by the Bible.]
During Second Temple era, we know of two main competing calendar systems. The "Jubilees" calendar, advocated by the Dead Sea sect consisted of twelve 30-day months with four extra quarterly days, adding up to 364. Because the number is divisible by 7, holidays would occur on the same day of the week every year--and never on the Sabbath.
The Pharisees and Rabbis observed a version of the Babylonian calendar: Months are defined by actual phases of moon: Approximately 29 1/2 days after previous new moon, adding up to 354 days.
Originally the New Moon was determined by actual testimony of witnesses to court (Sanhedrin). Under the Patriarch Hillel II (mid-4th century) a calculated calendar was introduced.
Because most holidays relate to agricultural seasons [e.g., Passover must be in Spring; Tabernacles at the ingathering time], an extra month is added periodically in order to correlate the 354-day lunar year with the 365 1/4-day solar year. The extra month is added after the last month [Adar].
"First month" = Nissan (Spring, time of Passover). But Jewish New Year is in the Seventh Month [=Tishri].
Days usually begin at sunset and end at nightfall.
Many of the holidays have dual themes-- natural / agricultural as well as historical.