View a video of the first night of the 2008 prayers here, from inside the Great Mosque in Mecca.
Shi'i Muslims do not recite the prayers and believe them to be an innovation from the time of the reign of 'Umar ibn al-Khattab, the second caliph (successor of the Prophet Muhammad.) The debate over this issue is too historical and complicated, and thus I will not delve into it here.
A summary of the general "Sunni" view can be found here. Ahadith (historical traditions reportedly connected to the Prophet Muhammad) from Sahih al-Bukhari, one of the major Sunni collections of such traditions, can be found here.
A rather terse summary of the Twelver Shi'i objections to Tarawih can be found here. Twelver Shi'is also reject the validity of Sahih al-Bukhari and the other major Sunni collections of Ahadith. They have their own collections, which include not only Prophetic traditions but traditions from their 12 Imams, religious-political leaders who lived during the seventh to tenth centuries.
A Brief Note on the Twelfth Imam, the Mahdi
The twelfth Imam is believed to have gone into a mystical occultation, from which he will return at a time appointed by God. He will usher in a period of absolute justice for his followers before the Day of Judgment. Presently, he is believed to be in a "greater" occultation (ghayba al-kubra), which began in 941 C.E. upon the death of Abu'l Hassan 'Ali ibn Muhammad al-Samarri ("the Samarran"), the last of four deputies (النوّاب) who they believe represented him for decades when he was in a "lesser" occultation (ghayba al-sughra). Twelver Shi'is believe he was sent into these occultations to protect him from his Earthly enemies.
Sunni and non-Twelver Shi'i Muslims do not believe in this line of 12 Imams. Isma'ili and Zaydi Shi'is break away with their own lines during the eighth century. Non-Shi'i/non-Muslim scholars believe that the Twelver belief in the occulted "hidden Imam" was the result of historical pressures, which in turn fueled messianic theological beliefs, in a way similar though not exactly the same as what happened in early Christianity (see Bart D. Ehrman's excellent book Lost Christianities.) Sunnis and other Shi'is have their own Mahdi figures. Jesus is also believed by all Muslims to return at the end of time.