An-Nawawi (may Allah have mercy on him) said: Taraweeh is Sunnah according to scholarly consensus.
End quote from Sharh al-Muhadhdhab, 3/526
Al-Khateeb ash-Sharbeeni (may Allah have mercy on him) said: They (the scholars) are unanimously agreed that it is Sunnah and that it is what is referred to in the words of the Prophet (blessings and peace of Allah be upon him): “Whoever prays qiyaam in Ramadan out of faith and seeking reward with Allah, his previous and later sins will be forgiven.”
Narrated by al-Bukhaari.
End quote from Mughni al-Muhtaaj, 1/460. For information on its being prescribed to offer it in congregation, please see the answer to question no. 21740.
With regard to ‘Umar (may Allah be pleased with him) being the one who first brought the people together to pray Taraweeh in congregation, whereas Abu Bakr (may Allah be pleased with them) did not do so, this question was answered by Imam ash-Shaatibi (may Allah have mercy on him) in his book al-I‘tisaam, 1/194, in which he said: Abu Bakr (may Allah be pleased with him) did not do that for one of two reasons:
1. either because he thought that the people’s praying qiyaam at the end of the night, and what they were doing, was better in his opinion than bringing them together to pray behind an imam at the beginning of the night; this was mentioned by at-Tartooshi;
2. or it was because he (may Allah be pleased with him) had little time to examine such minor issues, when he was focused on dealing with the apostates and other issues that were more important than Taraweeh prayer. Then when Islam became well-established at the time of ‘Umar (may Allah be pleased with him), and he saw the people scattered in the mosque, as it says in the report, he [‘Umar] said: If I bring the people together to pray behind one reciter, that will be better. When he did that, he pointed out that their praying qiyaam at the end of the night was better. Then the salaf (early generations) agreed that this was valid and they approved of it, and the ummah cannot unanimously agree on misguidance. The scholars stated that consensus can only be on the basis of shar‘i evidence. End quote.
And Allah knows best.