The Amidah (, "standing") is one of the two main prayers of Judaism.  It has that name because people say it standing up.  Jews say it at every prayer service of the year.

It is also called Shemoneh Esrei (, "eighteen") because at first the weekday version of the prayer had eighteen blessings.  (It now has nineteen.) In the Talmud, it is called Tefilah ().  This simply means "prayer".  It has that name because it is so central to Judaism.

Blessings of the Amidah
The Amidah always has three sections.  
The first section has blessings of praise to God.  
The middle section is different between weekdays and Shabbat and holidays.  On weekdays it has blessings asking for God's help.  On Shabbat and holidays there is one blessing to thank God for Shabbat or the holiday.
The last section has blessings to thank God for everything He does for us.

First section: praise to God
Praises God as the God of Abraham, Isaac and Jacob
Praises God as having power over life and death
Praises God for being holy

Middle section (weekday): asking God for help
Asks God for the ability to think and understand
Asks God to help us follow the laws of His Torah
Asks God to forgive us for sins
Asks God to save us from trouble
Asks God to heal the sick
Asks God to bless produce and to provide us a living
Asks God to return Jewish exiles to the land of Israel
Asks God to give us fair judges on Earth
Asks God to punish heretics who then slander Jews
Asks God to support righteous people
Asks God to rebuild the Temple in Jerusalem
Asks God to bring the Messiah
Asks God to accept all our prayers

End section: thanking God
Thanks God for allowing our prayers
General blessing of thanks to God
Thanks God for peace

Saying the Amidah
The Amidah is said at least three times every day: at the morning, afternoon and evening prayer services.  A person should say it standing.  A person should say it loud enough to hear it.  But it should be quiet enough that other people cannot hear it.

At the morning and afternoon services, the Amidah is repeated out loud by the prayer leader.  When the Amidah is said out loud:
The third blessing includes a section called Kedushah, or "holiness", centered around Isaiah's vision of angels declaring God's holiness.
The eighteenth blessing includes a section called Modim d'Rabbanan, or "The Rabbis' Prayer of Thanks." The Talmud says that everyone must thank God himself or herself, even during public prayer.
The Amidah is not said out loud at night.

In Reform Judaism the Amidah is not said quietly.  It is said out loud at every prayer service.

Changes on Shabbat and Jewish holidays
On Shabbat and important Jewish holidays, the Amidah prayer changes in two important ways.
People do not ask God to help them meet their needs.  The middle section of the weekday Amidah is replaced by one blessing about the holiness of the Shabbat or holiday.  The first three blessings and last three blessings are the same as on weekdays.
The Amidah is said a fourth time.  This Amidah is called the musaf, or additional service.  The middle blessing is about the additional offering given at the Temple in Jerusalem on Shabbat and holidays.
There are two Amidah prayers during the year that are different from any others.  See the pages on these holidays for more information.
On Rosh Hashanah, the musaf Amidah has nine blessings instead of seven.
On Yom Kippur, a fifth Amidah, called Ne'ilah ("closing") is said at the very end of the day.

Related pages
Siddur

Reference
A siddur (Jewish prayer book) is always the best reference for information about Jewish prayers: 

. 

It is a very good English/Hebrew siddur to use for learning about prayers.

Notes

Judaism