You may not wash or hang up wet laundry on Shabbat (or Jewish festivals).  The halachot for drying laundry depend on whether you use a clothesline or a dryer:

Shabbat: Laundry: Clothesline
You may take down laundry on Shabbat only if it was dry before sunset on Friday, and only if you don't:

• Transfer the laundry from one halachic domain to another (hotza'a), or
• Give the impression that the laundry had been washed on Shabbat (mar'it ayin).

If laundry on a clothesline is still wet at sunset on Friday, the laundry is muktza and you may not take it down or use it during Shabbat. This is different from the case of a dryer.
Reason On the clothesline, there is no certainty that the laundry will dry during Shabbat (it might rain, it might be cold or cloudy...), so the person cannot have in mind that it will dry during Shabbat.

Shabbat: Laundry: Dryer
Laundry in a dryer (even if it was wet at sunset) that was turned on before sunset on Friday (or Jewish festivals) is not muktza, even if you do not intend to wear it.  You may remove the dry laundry from the dryer on Shabbat as long as no light goes on.

You may not wash or hang up wet laundry on Jewish festivals (or Shabbat).  The halachot for drying laundry depend on whether you use a clothesline or a dryer:
Jewish Festivals: Laundry: Clothesline
You may only take down laundry on Jewish festivals if it was dry before sunset at the start of the festival, and only if you don't:

• Transfer the laundry from one halachic domain to another (hotza'a), or
• Give the impression that the laundry had been washed on the Jewish festival (mar'it ayin).

If laundry on a clothesline is still wet at sunset before the festival, the laundry is muktza and you may not take it down or use it during the festival. This is different from the case of a dryer.
Reason On the clothes line, there is no certainty that the laundry will dry during the festival (it might rain, it might be cold or cloudy...), so the person may not have in mind that it will dry during the festival.
Jewish Festivals: Laundry: Dryer
Laundry in a dryer (even if it was wet at sunset) that was turned on before sunset beginning the Jewish festival (or Shabbat) is not muktza, even if you do not intend to wear it.  You may remove the dry laundry from the dryer on the Jewish festival as long as no light goes on.

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).