Taraweeh, Salat al-Qiyam, Salat al-Lail and Salat al-Tahajjud?

The night prayer is any voluntary prayer performed at night, i.e. between Isha prayer and dawn. So, Al Tarawih, Al Tahajjud, Al Qiyyam are all from the night prayers. Al Tahajjud is the prayer performed following a period of sleep after which the person awakens especially for it which is also called Nashia al-Lail. See 6:73). Aisha ( may  Allaah  be  pleased  with  her) said: “Al Nashia (Al Tahajjud) is one performed after rising by night”. The night prayer is a confirmed Sunnah. But it is more insisted on during Ramadhan for the Hadith that the Prophet said: “Whoever prays during the nights of Ramadhan [Tarawih] with a firm belief and hoping for reward, all of his previous sins would be forgiven.”
Praying Tarawih just after Isha or later on is considered as Qiyyam in Ramadhan. But it is better to delay the Tarawih until the last third of the night – if this is not difficult for the Jamaa (Muslims) – for the following narration in Sahih Bukhari from ‘Abdur Rahman bin ‘Abdul Qari who said, “One night during Ramadhan, I went with ‘Umar to the mosque and the people were praying in different groups. Some were praying by themselves and others were praying in small groups. ‘Umar said: ‘I think it would be better if I gathered them under one imam.’ Then he did so and appointed Ubayy ibn Ka’b as the leader of the prayer. Then I went out with him on another night and all the people were praying behind one imam and ‘Umar said: ‘What a good innovation (bid’ah) this is,’ but, it is better to sleep and delay it until the latter portion of the night.” ‘Umar remarked, the prayer which they do not perform, but sleep at its time is better than the one they are offering.’ He meant the prayer in the last part of the night. (In those days) people used to pray in the early part of the night.”
As for the number of Tarawih Rakaa see Fatwa: 82507 and 82508.
Allah knows best.