What is the holy day of Judaism?

Answer 1

Judaism has many holy days. Shabbat (the sabbath), which begins at sundown on Friday and ends after twilight on Saturday, is the holiest.

Answer 2:

Many of these holidays are found in Leviticus ch.23. The holidays begin at sunset and last until after nightfall around 25 hours later.

• Shabbat - every Saturday (beginning Friday at sunset)
• Rosh Chodesh - the new moon, every 29 or 30 days
• Rosh Hashanah - the Jewish New Year, 2 days
• Aseret Yemei Teshuva - Ten Days of Repentance
• Yom Kippur - fast day, Day of Atonement, 1 day
• Sukkot - Tabernacles - 7 or 8 days
• Shemini Atzeret and Simchat Torah - 1 or 2 days
• Hanukkah - Festival of Lights - 8 days
• Tu Bishvat - New Year of the Trees - 1 day
• Purim - 1 day, followed by 1 day of Shushan Purim
• Pesach - Passover - 7 or 8 days
• Sefirah - Counting of the Omer - 49 days
• Lag Ba'omer - 1 day
• Shavuot - Feast of Weeks; Yom HaBikurim; Pentecost - 2 days
• The Three Weeks and the Nine Days (days of mourning preceding Tisha b'Av; see below)
• Tu B'Av - 1 day

Answer 2

Many of these holidays are found in Leviticus ch.23. Other occasions are more recent; specifically Purim (2375 years), Hanukkah (2200 years), and the fasts marking the Destruction of the Temple (Zechariah 7:3 and 8:19).The holidays begin at sunset and last until after nightfall around 25 hours later. They serve to enrich the Jewish year and to connect the people with their past.
All of these days are marked by added prayers and Torah-readings; and each has its specific observances.

• Shabbat - every Saturday (from Friday at sunset until Saturday after twilight)

Link: More about Shabbat

• Rosh Hashanah - the Jewish New Year, 2 days

Link: More about Rosh Hashanah

• Yom Kippur - a fast day, the Day of Atonement, 1 day

Link: More about Yom Kippur

• Pesach - Passover - 7 or 8 days

Link: Passover and the Seder

• Shavuot - Feast of Weeks; Yom HaBikurim - 1 or 2 days
• Sukkot - Feast of Booths - 7 days

Link: More about Sukkot

• Shemini Atzeret and Simchat Torah - 1 or 2 days

Minor holidays and occasions (in which work is not forbidden):

• Rosh Chodesh - the new moon, every 29 or 30 days

Link: The Hebrew calendar

• Hanukkah - the Festival of Lights - 8 days

Link: The founding of Hanukkah

• Tu Bishvat - New Year of the Trees - 1 day
• Purim - 1 day, followed by 1 day of Shushan Purim

Link: Purim and Queen Esther

• Sefirah - Counting of the Omer - 49 days
• Lag Ba'omer - 1 day

Link: What is Lag Ba'omer

• The Three Weeks and the Nine Days (days of mourning preceding Tisha b'Av; see below)
• Tu B'Av - 1 day

Fast days:
Judaism has six yearly fasts. The fasts start shortly before dawn and end at twilight, except for Yom Kippur and Tisha B'Av which start the evening before at sunset and last for 25 hours.

• Tzom Gedalya; the day after Rosh Hashanah
• Asara B'Tevet - 10th of the month of Tevet
• Shiva Asar B'Tamuz - 17th of Tamuz
• Tisha B'Av - 9th of the month of Av
• Ta'anit Esther - the day before Purim
• Yom Kippur

Four of the above fasts are in mourning for various stages in the destruction of the Temple, and are mentioned in Zechariah 8:19.

Link: The destruction
The Fast of Esther commemorates the danger that the Jews were in, during the events described in the Book of Esther.

The sixth fast, Yom Kippur, is the Day of Atonement, commanded in Leviticus 23:26-32.

Each festival has its specific purpose and laws:

• Rosh Chodesh marks the beginning of each Hebrew month (all of which are lunar) and is a minor holiday.
  
• On Rosh Hashanah, the shofar (ram's horn) is blown, to mark the beginning of the Jewish year.

• Yom Kippur is a fast day on which Jews pray for forgiveness for all their sins. No eating, drinking, or bathing is allowed. Wearing leather shoes is also prohibited.
• On Pesach, leavened bread, cakes, pasta etc. are forbidden; and unleavened Matzah is eaten. Passover begins with the Seder-meal, commemorating and retelling the story of the Exodus. Matzah and ceremonial foods are eaten at the Seder.

• On Shavuot the custom is to stay up all night studying Torah to mark the date that God gave the Ten Commandments.
• Sefirah - In the weeks between Pesach and Shavuot, we count the Omer, symbolizing the anticipation we felt in the days leading up to the Revelation at Mount Sinai. These weeks are a time of introspection and improvement.
• On Sukkot, Jews eat all their meals in outdoor arbor-canopied booths (Sukkah) in order to commemorate the Israelites' wanderings in the desert. Some will also sleep in the Sukkah. During the morning prayers on these days, we take the 4 minim consisting of a Lulav (young palm branch), an Etrog (Citrus Medica; citron), three Haddassim (myrtle branches) and two Aravot (willow branches).
• Shemini Atzeret and Simchat Torah - In Israel, these two occasions are observed on the same day (the eighth day from the beginning of Sukkot), while elsewhere they are kept separately (on the eighth and ninth days). This is a time of great rejoicing, with Shemini Atzeret symbolizing our close relationship with God (Rashi commentary, Numbers 29:35-36), and Simchat Torah celebrating the completion of the yearly cycle of reading the entire Torah scroll.

Link: More about Torah-scrolls

• On Hanukkah the 8-branched menorah is lit in the home; on the first night one candle, on the 2nd night 2 candles, until all 8 candles are lit on the 8th night.

• On Purim the story of Esther is read from a Megillat-Esther scroll and food baskets are given to friends and charity to the poor.

Reasons for the holidays:

Every one of them has as its purpose "remembering the Exodus from Egypt" (as stated in our prayers and the kiddush over wine). In addition, Passover is a Thanksgiving to God for the barley-harvest, Shavuot is a thanksgiving to God for the wheat-harvest, and Sukkot is a thanksgiving to God for the ingathering of grain.

Shavuot also celebrates the Giving of the Torah on Mount Sinai, and Sukkot commemorates God having protected us in the wilderness.

It may also be noted that it is instinctive and a moral and emotional need to celebrate in front of God every so often. This was Cain's motivation in making his offering in Genesis ch.4 without having been commanded.

Had God not given us the Torah-festivals listed above, we might instinctively seek out those of the Canaanites, which the Torah warns against (Exodus 34:15) immediately before listing the Jewish festivals (in the following verses).