Is it permissible to perform Tarawih prayer at home, and is it permissible to do so with one's wife, with the husband leading the prayer?

Tarawih prayer is a confirmed Sunnah (definition). The Prophet, peace and blessings be upon him, encouraged it by saying: “Whoever prays during the nights of Ramadan (definition) with faith and expecting Allah’s reward, all his previous sins will be forgiven.” The Prophet, peace and blessings be upon him, prayed it with his companions for several nights, then left it, fearing that it might be made obligatory for them. Umar ibn al-Khattab, may Allah be pleased with him, gathered them behind one imam.

It is preferable to pray Tarawih in the mosque in congregation. Scholars have agreed upon its desirability to be performed in congregation in the mosque, due to the hadith (definition): "Whoever prays with the Imam until he departs, it is recorded for him as spending the whole night in prayer," and because it is among the visible rituals. However, if a man prays Tarawih in his home individually or in congregation with his family, that is permissible.