According to many many Islamic schools that "tarawih" prayer is sunna and not fardh. I agree with that.
But it is there, not that it's not existed. I have got some friends who do not believe in "tarawih" prayer, they say that it's not from sunna, not written in Quran or Hadith. They follow the "shi3a" school. I am not attacking any of the Islamic madhahib, but I just need to know where there are some contradictions about this prayer. Your answers will be most apperciated! Thanks :)

Tarawih prayer is part of "sunna". So muslims are requested to pray it during Ramadan.

Below some details about this issue:

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RAMADAN NIGHTLY PRAYER (TARAWEEH)
Allah (SWT) has mandated to His servants acts of worship and made them of various kinds so as to provide them with means to pick and choose, and that others will not be bored with one act. Thus, we seek nearness to Him, through mandatory and non-mandatory deeds. Among the non-mandatory deeds, or Sunnahs, is the establishment of night time prayers (salatul lail), and Allah has praised those who observe them.

"Those who spend the night in adoration of their Lord, prostrate and standing." (Al-Qur'an 25:64)

"Their limbs do forsake their beds of sleep, the while they call on their Lord in fear and hope and they spend (in charity) out of the sustenance which we have bestowed on them." (Al-Qur'an 32:16)

These two verses indicate the basis for nightly prayer in Al-Qur'an and the good return awaiting those who observe it. In a hadith, the Messenger of Allah (saas) says: "The best prayer after the obligatory ones is the night prayer." (Muslim)

Elsewhere he says: "O people! disseminate the salutations of peace As-Salaam 'Alaikum; feed the needy food, and join the blood ties among the next of kin; and observe night prayer while people are at sleep, you will enter paradise peacefully." (Tirmidhi)

One of the night prayers, is Witr, the odd numbered bedtime prayer. It should be the last prayer said before retiring. The minimum Witr is one rak'ah and the maximum is eleven raka'ats. The Prophet (saas) said: "Whoever wants to pray Witr with one rak'ah should do it and whoever wants to pray Witr with three raka'ats should do it." (Abu Dawud/Nasa'e).

However, the Messenger of Allah (saas) was always consistent with eleven raka'ats for night prayers, according to his wife, 'Aishah (raa) who reported: "The Prophet never observed prayers after `Isha (night obligatory prayer) and before Fajr (morning prayer), more than eleven raka'ats. He saluted after every two raka'ats, and he prayed Witr with one raka'at". (Jama'ah, except Tirmidhi)

In a different procedure, one may make four raka'ats, and salute, and continue with another four and salute, then he caps them with three raka'ats of Witr. 'Aishah (raa) reported: "The Prophet used to pray four raka'ats and one cannot describe their beauty, and their length, (he salutes) then makes another set of four raka'ats, one cannot describe their beauty and length, he then caps it with three raka'ats." (Agreed upon)