Is Tarawih Prayer Bid`ah?
Question

On the occasion of the blessed month of Ramadan, the people turn to Tarawih prayer. My question is: there are some who pray eleven rak’ahs immediately after ‘Isha’ prayer, following the example of the Messenger of Allah (peace and blessings of Allah be upon him); and there are those who pray twenty-one rak’ahs, ten after ‘Isha’ and ten before Fajr, then they pray Witr. What is the Shar’i ruling on this way of praying Tarawih? Please note that there are those who say that praying Qiyam before Fajr is Did’ah. May Allah reward you with good.
Praise be to Allah.Is Tarawih Prayer Bid`ah?
Tarawih prayer is Sunnah according to the consensus of the Muslims, as Al-Nawawi (may Allah have mercy on him) said in Al-Majmu’. 
Tarawih prayer was encouraged by the Prophet (peace and blessings of Allah be upon him), for example when he said: “Whoever spends the nights of Ramadan in prayer out of faith and in the hope of reward, his previous sins will be forgiven.” Narrated by Al-Bukhari, 37; Muslim, 760 
So how can Tarawih prayer be a Bid`ah when the Prophet (peace and blessings of Allah be upon him) encouraged us to do it and the Muslims are unanimously agreed that it is recommended!? 
Is gathering to offer Tarawih prayer in the mosques Bid`ah?
Perhaps the one who said that Tarwaih prayer is Bid`ah meant that gathering to offer Tarawih prayer in the mosques is Bid`ah. 
This too is incorrect, because the Prophet (peace and blessings of Allah be upon him) prayed Tarawih in congregation with his companions for many nights, then he stopped doing that lest it be made obligatory upon the Muslims. Then when the Prophet (peace and blessings of Allah be upon him) died and the revelation came to an end, this concern was no longer an issue, because it could not be made obligatory after the death of the Prophet (peace and blessings of Allah be upon him). So `Umar (may Allah be pleased with him) gathered the Muslims to pray Tarawih in congregation .
The time for Tarawih prayer is from after `Isha’ prayer until dawn comes. 
Number of Rak`ahs in Tarawih prayer
There is no specific number of Rak`ahs for Tarawih prayer , rather it is permissible to do a little or a lot. The two ways mentioned in the question are both permissible. 
That should be based on whatever the people in each mosque think is suitable for them. 
However, the best is that which was proven from the Prophet (peace and blessings of Allah be upon him), which is that he never prayed more than eleven Rak’ahs when praying Qiyam Al-layl, in Ramadan or at other times. 
Shaykh Ibn ‘Uthaymin said, after mentioning the number of Rak'ahs in Tarawih prayer:
“The matter is broad in scope. No one should be denounced for praying eleven or twenty-three, because the matter is broader in scope than that, praise be to Allah.” (Fatawa Al-Shaykh Ibn ‘Uthaymin, 1/407) 
And Allah knows best.