Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
There is a general and specific relationship between Taraweeh prayer and Qiyam al-Layl (Night Prayer); thus, Taraweeh is considered a form of Qiyam al-Layl, but not every Qiyam al-Layl is considered Taraweeh.
Qiyam al-Layl and Taraweeh share the fact that both are recommended Sunnah acts (Mandub) and both are performed during the night.
However, they differ in several matters, including:
The term "Taraweeh" is applied specifically to Qiyam al-Layl in Ramadan; therefore, it is a standing (Qiyam) specific to Ramadan. As for standing in prayer outside of Ramadan, it is considered general voluntary prayer (Nafl Mutlaq).
The time for Taraweeh prayer begins after performing the 'Isha prayer, whereas the time for Qiyam al-Layl begins from Maghrib.
It is recommended to perform Taraweeh in congregation, unlike the general Qiyam prayer, for which congregation is not specifically recommended.
Taraweeh is prayed with the intention of the "Ramadan Standing" (Qiyam Ramadan) or the intention of "Taraweeh," whereas Qiyam is valid with a general intention.
It is recommended for Taraweeh to have a specific number of Rak‘ahs; it is prayed as twenty Rak‘ahs for completeness. Ibn Abi Shaybah narrated in his Musannaf from Ibn 'Abbas (may Allah be pleased with him): "The Messenger of Allah ﷺ used to pray twenty Rak‘ahs and the Witr in Ramadan." And in Al-Sunan al-Kubra by Al-Bayhaqi, from Ibn 'Abbas (may Allah be pleased with him): "The Prophet ﷺ used to pray twenty Rak‘ahs and the Witr in the month of Ramadan outside of congregation."
In Taraweeh, one does not combine more than two Rak‘ahs with a single Salam. If one prays four, for example, with one Salam, it ceases to be Taraweeh because it resembles the obligatory (Fard) prayer. Sheikh al-Islam Imam Muhammad al-Ramli said: "If he prays four with one Tasleem, it is not valid if he did so intentionally and knowingly; otherwise, it becomes a general voluntary prayer because it contradicts what is prescribed" [Nihayat al-Muhtaj, Vol.2/P.127]. This is unlike Qiyam, where it is permissible to pray two, four, or whatever one wishes [with one Salam].
It is better to perform Qiyam al-Layl at home, unlike Taraweeh prayer, for which it is better to gather in the mosque.
As for Tahajjud, it differs from Qiyam al-Layl and Taraweeh in that it occurs specifically after waking up from sleep at night.
And Allah the Exalted knows best.
