When the month of Ramadan begins, Muslims enter into a period of discipline and worship: fasting during the day, and praying throughout the day and night. During Ramadan, special evening prayers are conducted during which long portions of the Qur'an are recited. These special prayers are known as taraweeh.
The word taraweeh comes from an Arabic word which means to rest and relax. The prayer can be very long (well over an hour), during which one stands upright to read from the Qur'an and performs many cycles of movement (standing, bowing, prostrating, sitting). After each four cycles, one sits for a brief period of rest before continuing -- this is where the name taraweeh ("rest prayer") comes from.
During the standing portions of the prayer, long sections of the Qur'an are read. The Qur'an is divided into equal parts (called juz) for the purpose of reading sections of equal length during each of the Ramadan nights. Thus, 1/30 of the Qur'an is read on successive evenings, so that by the end of the month the entire Qur'an has been completed.
It is recommended that Muslims attend the taraweeh prayers in the mosque (after 'isha, the last evening prayer), to pray in congregation. This is true for both men and women. However, one may also perform the prayers individually at home. These prayers are voluntary, but are strongly recommended and widely practiced.
There has been some dispute about how long the taraweeh prayer is supposed to be: 8 or 20 raka'at (cycles of prayer). It is without dispute, however, that when praying the taraweeh prayer in congregation, one should start and end with the imam, according to the number that he performs. Night prayers in Ramadan are a blessing, and one should not argue about this fine point. For more details about the evidence in support of each opinion, please see the articles linked below.
Saudi Arabia television broadcasts the taraweeh prayers live from Mecca, Saudi Arabia, now with simultaneous subtitling of the English translation. See the links below to watch.