The Hillel II calendar (הַלּוּחַ הָעִבְרִי) is the fixed, calculated Hebrew calendar established around 358–359 CE by the Sanhedrin under Hillel II. Before this, the new month was declared by the Sanhedrin based on eyewitness reports of the new crescent moon.
The calendar is lunisolar — months follow the lunar cycle (~29.5 days each), while the year stays aligned with the solar year through a 19-year Metonic cycle with 7 leap years (שנת עיבור) that add a 13th month (Adar II).
The Hebrew year begins with Tishrei (the civil new year, Rosh Hashanah) in autumn. Religiously, Nisan is the "first month" (as commanded in Exodus 12:2). The months in order from Tishrei are:
Tishrei (30) · Cheshvan (29/30) · Kislev (29/30) · Tevet (29) · Shevat (30) · Adar (29; or Adar I 30 + Adar II 29 in leap years) · Nisan (30) · Iyar (29) · Sivan (30) · Tammuz (29) · Av (30) · Elul (29)
The year has 353–355 days (regular) or 383–385 days (leap year).
Rosh Chodesh (ראש חודש, "head of the month") is the celebration of the new month. It occurs on the 1st of each Hebrew month, and also on the 30th of months that have 30 days (a two-day Rosh Chodesh).
Historically, it was declared when two witnesses reported seeing the new crescent moon from Jerusalem. Today, it follows the fixed calculated calendar.
The Sefirat HaOmer (ספירת העומר, "Counting of the Omer") is a 49-day period from the second night of Passover (16 Nisan) through the day before Shavuot. Each of the 49 days corresponds to one of 49 combinations of the 7 lower Sefirot (divine attributes): Chesed, Gevurah, Tiferet, Netzach, Hod, Yesod, and Malchut.
The fiftieth day is Shavuot, commemorating the giving of the Torah at Mount Sinai.
The Torah commands three pilgrimage festivals (שָׁלֹשׁ רְגָלִים):
Pesach / Passover (15–22 Nisan) — celebrating the Exodus from Egypt and the spring harvest.
Shavuot / Feast of Weeks (6–7 Sivan) — celebrating the giving of the Torah at Sinai and the wheat harvest. Falls 50 days after Passover.
Sukkot / Feast of Tabernacles (15–22 Tishrei) — celebrating the 40 years of wandering in the wilderness, with harvest thanksgiving.
Additionally, Rosh Hashanah (1–2 Tishrei) is the Jewish New Year and Yom Kippur (10 Tishrei) is the Day of Atonement — the holiest day of the year.
The Hebrew day begins at nightfall (tzet hakochavim) — when three stars become visible, approximately 50 minutes after sunset (or when the sun is 8.5° below the horizon). This follows Genesis 1:5: "And there was evening, and there was morning — the first day."
This site accounts for this: after sunset in Jerusalem, the Hebrew date shown advances to the next day.
Candle lighting is shown as 18 minutes before sunset on Friday evening, following the common Ashkenazi custom. Many communities use 20 minutes; Jerusalem uses 40 minutes.
Havdalah (the ceremony ending Shabbat) is shown for when the sun's center reaches 8.5° below the horizon on Saturday night — the standard tzet hakochavim used by most halachic authorities.
Times are calculated for Jerusalem using precise astronomical calculations via PyEphem.