Firstly i would like to point out that "qiyam" means to wake or to get up from sleep, so qiyam prayer would be prayer that you wake from sleep and pray. So tahajjud prayer, witr and any other prayer that is prayed after awakening from sleep would be considered qiyam.
Procedure
prophet muhammad saaws used to pray an odd number of rakaat it would be two rakaat at a time and in addition would be witr, one rakaat so it would come out to be an odd number. Each of his rakaat would get shorter as he continued making the first the longest and the last the shortest.
Narrated 'Abdullah bin 'Umar : A man said, "O Allah's Apostle! How is the prayer of the night?" He said, "Two Rakat followed by two Rakat and so on, and when you apprehend the approaching dawn, offer one Raka as Witr." bukhari 238, 21
Yahya related to me from Malik from 'Abdullah ibn Abi Bakr from his father that 'Abdullah ibn Qays ibn Makhrama told him that Zayd ibn Khalid al-Juhani said one night that he had gone to observe the prayer of the Messenger of Allah, may Allah bless him and grant him peace. He said, "I rested my head on his threshold. The Messenger of Allah, may Allah bless him and grant him peace, got up and prayed two long, long, long rak'ats. Then he prayed two rak'ats which were slightly less long than the two before them. Then he prayed two rak'ats which were slightly less long than the two before them. Then he prayed two rak'ats which were slightly less long than the two before them. Then he prayed two rak'ats which were slightly less long than the two before them. Then he prayed two rak'ats which were slightly less long than the two before them. Then he prayed an odd rak'at, making thirteen rak'ats in all."muslim
Amount of rakaat
The maximum of rakaat would be 11, excluding two rakaat sunnah of fajr, or 13 with the sunnah prayer.
Narrated 'Aisha, The Prophet (p.b.u.h) used to offer thirteen Rakat of the night prayer and that included the Witr and two Rakat (Sunna) of the Fajr prayer. bukhari 241, 21
When to pray
The prophet would wake from his sleep at the lattest part of the night before fajr, when he finished praying his 13 rakaat he would go lie down untill the athan was pronounced then he would go to the mosque to pray fajr
Narrated Al-Aswad: I asked 'Aisha "How is the night prayer of the Prophet?" She replied, "He used to sleep early at night, and get up in its last part to pray, and then return to his bed. When the Muadh-dhin pronounced the Adhan, he would get up. If he was in need of a bath he would take it; otherwise he would perform ablution and then go out (for the prayer)." bukhari 247,21
From the quran
Allah subahanahu wa ta3ala has said in the quran that those who regularly perform the tahajjud prayers are Righteous and are more deserving of Allah's bounty and mercy. Allah says:
"Lo! Those who keep from evil will dwell amid gardens and watersprings, taking that which their Lord gives them. For, lo, they were doers of good. They used to sleep but little of the night and before the dawning of each day would seek forgiveness.''
Allah praised and complemented the deeds of those who perform the late-night prayers.
"The slaves of the Beneficent are they who walk upon the earth in humbleness, and when the ignorant address them, they say: 'Peace' and they who spend the night prostrating before their Lord and standing" [ al-Furqan: 63-64 ].
So in addition to us following the prophet by praying the night prayer, we will be pleasing our lord and this is the reason why the prophet encouraged us to awaken from our sleep and pray to Allah.
