Narrated Abu Huraira: Allah's Messenger (ﷺ) (p.b.u.h) said, "Our Lord, the Blessed, the Superior, comes every night down on the nearest Heaven to us when the last third of the night remains, saying: "Is there anyone to invoke Me, so that I may respond to invocation? Is there anyone to ask Me, so that I may grant him his request? Is there anyone seeking My forgiveness, so that I may forgive him?"(Sahih Bukhari)
Tarawih, Tahajjud, and Qiyaamul Layl are all names used for the night prayer. (when it is in Ramadan, it is called Tarawih).
The Superiority of the Night Prayer (Tahajjud) and Its Effect on Eemaan (PDF, 5 pages)
When Do We Pray the Night Prayer?
How Do We Make the Night Prayer?
This page discusses different ways (i.e. number of rakahs and manner of making them) that the Prophet (sallAllahu alayhi wa sallam) made the night prayer.
Performance of the Night Prayer - Shaykh Albaani
This also discusses the different number of rakaat and how they are to be made. It can also be found here.
More on the Manner of Making the Night Prayer
Kitaabul Tahajjud (Book of Prayer at Night, from Sahih Bukhari) - English and Arabic
The Night Prayer During Ramadan
The night prayer during Ramadan is called the Tarawih prayer (Salaatul Tarawih)
The Salaf and Their Standing For The Night Prayer - Abu Idress Muhammad (audio, 15:55) - Also includes brief notes from the lecture.
Related
Night Prayer Tag @Salaficentre.com - various posts related to the night prayer
The Night Prayer in Ramadan - Shaykh Albaani (the site is currently offline but I am leaving up the link for now)
The Night Prayer in Ramadan - Shaykh Albaani (PDF, Al Ibaanah)
