Tarawih prayer is Sunnah Mu'akkadah for men and women since the blessed Messenger said:

'The month in which Allah has prescribed for you fasting and I have initiated for you the night prayer.Whoever fasts in it and prays with sincerity and faith, sins leave him clean and the day his mother gave birth to him'. (Ibn Maja)

The Tarawih with congregation is more virtuous since `Umar (r) established it with the congregation and the congregational prayer is a communal Sunnah. If no one prayed it in the Mosque all the people have sinned and missed the Sunnah.

The intention for Tarawih is to intend to pray Tarawih or Sunnah of the time, or the night standing in Ramadan. And its time is after `Isha prayer and it is twenty rak`ats with ten salams. Yazîd ibn Rûman says: in the time of `Umar (ra) people used to pray twenty three rak`ats (inclusive of the three rak`ats of witr)." (Muwatta Imam Malik)

It is desirable to sit and rest for a while after every four rak`ats as Tarawih is derived from the word raha (to rest). One can remain silent if he wishes or recite La ilaha illallah or recite the Glorious Qur'an or send blessings upon the Messenger or pray nafl alone or read the tasbih:

"subHana dhil-mulki wal-malakût, subHana dhil-`izzati wal-`aZmati wal-haybati wal-qudrati wal-kibri-ya'i wal-jabarût, subhanal-malikil-Hayyil-ladhî la yanamu wa la yamût, subbûHun quddûsun rabbuna wa rabbul-mala'ikati war-rûH, Allahumma ajirna minannar, ya
mujîru ya mujîru ya mujîr"

Which means:

"Glory to Lord of power and Kingdom. Glory to the Noble and the Great, the One who possesses Majesty, Authority, Grandeur and Dominance. Glory to the King Who is living and neither sleeps nor dies. The Glorified, the Holy, our Lord and Lord of the angels and soul. O Lord save us from the hell fire O Saver, O Saver, O Saver."

It is Sunnah to complete the glorious Qur'an once in the month and if people find it difficult to cope with, then less should be read to avoid their annoyance. The recitation of blessings should not be left out in tashah-hud. Similarly the thana and tasbîhs in rukû' and sajdah must not be left out. There is no need to make up (qada) missed Tarawih since qada is only for fard and wajib.

Whoever prays `Isha alone it is permissible for him to pray Tarawih with the imam. If he has missed some rak`ats of Tarawih it is still permissible for him to pray witr with the congregation and then complete the Tarawih afterwards or alternatively he can complete the Tarawih and pray witr alone. The scholars differ on the priority of the two.

I say that it is better to pray witr with the congregation since congregational prayer is better than praying alone.
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Source:
M. I. H. Pirzada, Islamic way of Worship (al-Karam Publication, 2nd
edition,
1998), p 100-101.