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).

Men are required to hear at least 60 shofar blasts on Rosh Hashana (l'chatchila) in order to fulfill the commandment of hearing shofar:  30 before the musaf amida and 30 afterward.  But they fulfill their requirement (b'di'avad) if they have heard at least 30 on each day of Rosh Hashana. 
Women only need to hear 30 shofar blasts on each day of Rosh Hashana.

Teruma that is separated and destroyed is not considered to be a violation of bal tashchit, since it is done to fulfill a commandment/mitzva.

You may not separate teruma and ma'aser on Shabbat since it is “fixing” the food by making it usable.

Teruma gedola is 1/50th of the total food. But today, since the cohen does not eat it, we remove a smaller amount.

Separate teruma and ma'aser on any amount of produce grown in Eretz Yisrael.
 

Halachically ownerless (hefkeir) fruit is fruit that will not be collected or picked by or for the owner, whether the fruit is:

• Still on the tree or fallen on the ground.
• Growing on public grounds or privately owned property.

 

“Ownerless” (hefkeir) fruit does not require having teruma and ma'aser separated even after it has been brought into your house.

Separate teruma and ma'aser on herbs grown in Eretz Yisrael.

Teruma and ma'aser laws only apply to produce grown in Eretz Yisrael.

For more on teruma and ma'aser, including a short form of the text, see kashrus/kk-medi-terumos.htm" target="_blank">Star-K article

You may not put shmita peels, cores, and other waste parts into the garbage (unlike teruma, you may not double-bag them and put them in the garbage).  You must put them aside to rot before disposal.

All of the Jewish festival holidays had an agricultural element to them.
 
Agricultural laws include Kilayim, Orla, Reishit, Teruma/Ma'aser, Shmita, and Yashan, as well as special laws applying only to fruit trees. Some of these laws still apply today by Torah law (d'oraita) while others, such as First Fruits (bikurim), only apply when the Jerusalem Temple stands and so are not practiced now. Others are observed today as "practice" for when the Temple is rebuilt.

Spices (pure) are OK everywhere but beware of bugs. If the spices are ground, you may use them.