Can You Pray Tarawih at Home?
Question

Is it permissible to pray Tarawih at home? Is it permissible to pray it with one's wife, with the husband as the imam?
Praise be to Allah.Is tarawih obligatory?
Tarawih prayer is Sunnah mu’akkadah (a confirmed Sunnah) which the Prophet (peace and blessings of Allah be upon him) encouraged when he said: “Whoever prays at night in Ramadan out of faith and the hope of reward, his previous sins will be forgiven.” (Narrated by al-Bukhari, 37; Muslim, 759) 
The Prophet (peace and blessings of Allah be upon him) led his Companions in praying Tarawih for several nights, then he feared that it may be made obligatory for them, so he did not go out to them. Then `Umar united the people under one imam, and it has been prayed in congregation ever since. 
Isma’il ibn Ziyad said: “`Ali passed by the mosques in which there were lamps during the month of Ramadan and said, ‘May Allah illuminate `Umar’s grave for him as he illuminated our mosques for us.” (Narrated by al-Athram, quoted in al-Mughni, 1/457) 
Can you pray tarawih at home?
Al-Bahuti said in Daqaiq Uli al-Nuha, 1/2245: 
It is better to pray Tarawih in the mosque than at home, because the Prophet (peace and blessings of Allah be upon him) gathered the people to pray Tarawih for three consecutive nights, as was narrated by `Aishah… and he (peace and blessings of Allah be upon him) said: “Whoever prays qiyam with the imam until he leaves, it will be as if he spent the whole night in prayer.” 
Al-Shawkani said in Nayl al-Awtar (3/62): 
“Al-Nawawi said: The scholars agreed that it is mustahabb, but they differed as to whether it is better to pray it alone at home or in congregation in the mosque. Al-Shafi`i and a number of his companions, and Abu Hanifah, Ahmad, some of the Malikis and others said that it is better to pray it in congregation as `Umar ibn al-Khattab and the Companions (may Allah be pleased with him) did and as the Muslims continued to do, because this is one of the manifest rituals. 
So praying it in congregation in the mosque is better, but if a man prays it on his own at home, or in congregation with his family , that is permissible. 
Al-Nawawi said in al-Majmu’, 3/526: 
“Tarawih prayer is Sunnah according to scholarly consensus… it is permissible to pray it alone or in congregation , but which is better? There are two well-known points of view concerning this; the correct view according to the consensus of the Companions is that it is better to pray it in congregation.” 
For more, please see this category: Tarawih Prayer and Laylat al-Qadr.
And Allah knows best.