"I happened to pray one night with the Prophet, peace and blessings be upon him. The Prophet started reciting al-Baqarah chapter of the Quran and I thought he would stop after 100 verses.
But when he went beyond it I thought that he may want to recite the whole chapter in one Rakah.
When he finished al-Baqarah I thought he would do Ruku but then he immediately started reciting al-Imran and when he finished he started reciting an-Nisa.
The Prophet was reciting very slowly with enough pauses and would do Tasbih (praising God) and Dua (supplication) according to the subject being discussed in the relevant Ayah.
After that the Prophet did Ruku (bowomg). In Ruku he stayed as long as he did when he was in Qiyam (standing in prayer). After Ruku he stood up for almost same time and then he performed Sajdah (prostration) and stayed there as long as he recited Quran while doing Qiyam". (Hudaifa, may God be pleased with him, narrated this hadith as in Sahih al Muslim, Nasai)
Of course, not all the Prayers of the Prophet were this long. In public he would pray for a shorter period of time and ask other imams to do the same. The Prophet use to make dua in Sajdah not just tasbeeh as we do in obligatory prayers. He used to cry in Sajdah. He would spend an extended amount of his time in Sajdah whenever he was praying on his own. Many times, however, Muslims would join him when they found him praying alone.
Aisha, may God be pleased with her, mentions that: the Prophet one night stood up for Salah and he stayed in it for so long that I thought the Prophet had passed away or died. When I felt that way I stood up shook his toe and I felt the movement then I laid down again and I heard the Prophet saying in Sajdah "I seek refuge in Your pleasure from Your wrath, and in Your pardon from Your punishment, and in You from You. I cannot enumerate Your praises as You praise Yourself." (Transliteration: Audhu bi ridaka min sakhatika, wa bi muafatika min uqubatika wa bika minka, la uhsiy thana'an alayka, anta kama athnayta ala nafsika).
When he stood up from the Sajdah he asked Aisha, "do you think God's Prophet has betrayed you?" Aisha responded "No Preophet of God, because of the long Sajdah I thought you had died." (hadith from Baihaqi but Dua wording from Muwatta Imam Malik)
One of the Prophet's companion, Abdullah ibn Zubayr, would pray with such concentration that when he was in Sajdah the sparrows would come flying and sit on his back. In a separate narration, Abdullah ibn Abbas, another companion, said if you want to see how the Prophet of God used to pray, you should copy how Abdullah ibn Zubayr used to pray.
And consider these two Hadiths from the Sahih Muslim:
Ma'dan b. Talha reported: I met Thauban, the freed slave of God's Messenger, and asked him to tell me about an act for which, if I do it, God will admit me to Paradise, or I asked about the act which was loved most by God. He gave no reply. I again asked and he gave no reply. I asked him for the third time, and he said: I asked God's Messenger about that and he said: Make frequent prostrations before God, for you will not make one prostration without raising you a degree because of it, and removing a sin from you, because of it. Ma'dan said that then he met Abu al-Darda' and when he asked him, he received a reply similar to that given by Thauban.
In the second Hadith, Rabi'a b. Ka'b said: I was with God's Messenger one night and I brought him water and what he required. He said to me: Ask (anything you like). I said: I ask your company in Paradise. He (the Prophet) said: Or anything else besides it. I said: That is all (what I require). He said: Then help me to achieve this for you by devoting yourself often to prostration.
Sajdah is truly a humbling experience. We can beg for God's forgiveness and cry by thinking about our misdeeds, as well as seek refuge in God from the Hellfire. We are in one of the most submissive physical positions when in Sajdah. It is one of the best occasions to ask God for forgiveness, guidance, and all that we want. It is one of the best positions in which to talk to God. Seeking God's pleasure and forgiveness need to be given top priority in these remaining days and nights of Ramadan.