Jewish Festivals are listed in the Torah and are of two types: 

1. Three pilgrimage festivals (shalosh regalim):

• Passover,
• Shavuot, and
• Sukkot (including Shimini Atzeret).

These festivals were celebrated in ancient times by "appearing before God"--by bringing offerings to the Tabernacle or Temple.

2. High Holidays: 

• The Jewish New Year (Rosh Hashana—“Yom Teru'a” in the Torah), and the
• Day of Atonement (Yom Kippur).

 
Duration
Jewish holidays that were originally one day are now observed as follows:

• Rosh Hashana--2 days.
• Yom Kippur--1 day.
• Shavuot--1 day in Eretz Yisrael or 2 days elsewhere.
• Passover has festival days at the beginning and end of the holiday and intermediary days of lesser holiness, which are called chol ha'moed. In Eretz Yisrael, Passover lasts for 7 days in total and the first and last days are festival days. Outside of Eretz Yisrael, Passover lasts 8 days and has two festival days at the beginning and two at the end.
• Sukkot has festival days at the beginning and at the end and intermediary days of lesser holiness, which are called chol ha'moed. In Eretz Yisrael, Sukkot lasts for 8 days and the first and last days are festival days (the last day is Shimini Atzeret). Outside of Eretz Yisrael, Sukkot lasts for 9 days and the first two days and last days are festival days (the 8th day is Shimini Atzeret and the 9th day is Simchat Torah).

Character
Each holiday contributes its own character to Jewish life (Passover--the theme of freedom; Yom Kippur brings atonement, etc.). 

Celebrating
How to celebrate these holidays is detailed in our Oral Law and halacha books. Jewish festivals as practiced today are similar in holiness to Shabbat. As with Shabbat, the Jewish festival has candle lighting, kiddush at two meals, and havdala.   We eat our best food and wear our best clothing on Jewish festivals (we eat our next-best food and wear our next-best clothing on Shabbat!).
The main idea behind eating meals on Jewish festivals is joy (simcha), so you should drink wine and eat meat (only if you enjoy wine and meat).  There is no third meal on Jewish festivals since people used to eat two meals each day (adding a third meal on Shabbat was for enjoyment/oneg).
 
Melachot
Any activities or actions permitted on Shabbat are also permitted on the Jewish festivals. Actions that are forbidden on Shabbat are generally also forbidden on Jewish festivals, but there are some leniencies (only if the actions are needed for that festival day).
Examples

• Lighting from an existing flame,
• Cooking and baking for the Jewish festival day, and
• Carrying outside the eruv (hotza'a--transferring objects between domains).

 
Grama
Grama (indirectly causing an action) is permitted on Jewish festivals (but not on Shabbat). For example, you may advance or delay a timer that will make a light go on or off in the future (the timer must already be plugged in and operating from before sunset of the festival day).
Note For an action to be considered indirect based on time, there must be at least 2.5 seconds after the first action is done before the resulting action begins to happen.

Psik Reisha Dla Neicha Lei
Psik reisha dla neicha lei is forbidden on Jewish festivals, just as it is on Shabbat.

D'oraita Restrictions
D'oraita restrictions apply world-wide to:

• First and seventh days of Passover,
• First and eighth days of Sukkot,
• First day of Shavuot,
• Yom Kippur,
• First day of Rosh Hashana.

Note The same restrictions apply to all other Jewish festival days but are rabbinical.
 
Women
In general, women are not required to perform the positive, time-dependent commandments. Women and girls are not required to eat any Jewish festival meals except the Passover seder meal (but they are not allowed to fast on those days).

For greetings for the New Year, say:

• “Ketiva V'Chatima Tova”

              From: Rosh Chodesh Elul
              Until:  Eve of (erev) Rosh Hashana (when the holiday begins).

• “L'Shana Tova Tikateiv V'Tichateim”

            From:  First night of Rosh Hashana (when the holiday begins)
            Until:   Musaf of the first day of Rosh Hashana.

• “Gmar Chatima Tova”

               From:  Musaf Rosh Hashana
               Until:   End of Yom Kippur.

• “Gmar Tov”

               From: Yom Kippur
               Until:  Musaf Hoshana Rabba.
 

Saying Baruch ha'mavdil bein kodesh l'chol after dark at the end of a Jewish festival or of Rosh Hashana does not affect the additions you will then say in birkat ha'mazon.

There are two main types of bowing during the prayer services: 

• Bowing just from the waist (with and without taking steps), and
• Bowing with the knees, plus two variations on Rosh Hashana and Yom Kippur (hands and knees on floor).

I. Waist-Bowing (Two Forms)

• Modim;
• Modim in reader's repetition of amida;
• Bar'chu;
• Lecha Dodi; and
• Alenu.

          To bow this way, keep your legs straight and bend forward from your waist.
    B.  Waist-Bowing/Take Steps, for:

• Amida, and
• Kaddish.

• Bow down from waist with your legs straight.
• Take three steps backward (left foot, right foot, left foot),
• Then:
  • Bow from your waist to the left and say, oseh shalom bi'mromav,
  • Bow from your waist to the right and say, hu ya'aseh shalom alenu, and
  • Bow from your waist to the front and say, ve'al kol yisrael v'imru amen.

II.    Knee-Bowing (Three Times during Amida)

• Beginning of amida's first blessing,
• End of amida's first blessing, and
• Next-to-last amida blessing: ha'tov shimcha.

• Bend knees (at baruch),
• Bow forward (at ata), and
• Straighten up (before God's name).

III.  Knee-Bowing to Floor (Two Forms)
       A.  Knee-Bowing to Floor--Hands and Knees Only

• Kneel (with your back straight up) (at “hayu kor'im”), and
• Bow down with hands and knees on floor (at “u'mishtachavim u'modim”), but
• You are not required to touch your forehead to ground.
  Note You should still bow from your waist (but not to the floor) on Rosh Hashana--even if you are praying alone and even if there is no Torah present.

• Kneel (with your back straight up) (at “hayu kor'im”),
• Bow down with hands and knees on floor (at “u'mishtachavim u'modim”),
• Touch forehead to floor (at v'noflim al pneihem).