Of the great mercy of Allaah سبحانه و تعالى is that he has favoured certain times over others. Thus, He has distinguished these timings so that the worshipper can eagerly anticipate their arrival, and thus pray earnestly and sincerely. Had all timings been the same, the worshipper would not have had the enthusiasm and fervor that exists during more blessed times.
Therefore is it important that the one who desires that his du'aa' be answered utilise these times, and ensure that his du'aa' be more frequent and sincere during them. One of these special times is during the last third of the night. The night extends from sunset to dawn, and can be divided into three parts. If, for instance, sunset is at 6:00 p.m., and dawn at 4:00 a.m., the night is then 10½ hours long, and each one third is 3½ hours long. The last one third would then extend from 1:00 a.m. to 4:30 a.m. 
During this time, when most of creation is in a deep sleep, the true worshipper is awake, earnestly praying to his Creator سبحانه و تعالى, reflecting upon Allaah's creation, the heavens and the earth. In fact, Allaah سبحانه و تعالى describes the true believers as those who: "And in the hours of dawn, they seek forgiveness from their Lord."
Not only that, but Allaah Himself سبحانه و تعالى comes down to His servant, and opens for Him the doors of Mercy and Forgiveness. Abu Hurayrah رضي الله عنه narrated that the Prophet صلى الله عليه وسلم said: "Our Lord descends every night, during the last third of it, to the skies of this world, and asks: 'Whos is making du'aa' to Me, so that I can respond to him? Who is asking Me, so that I can give him? Who is asking My forgiveness, so that I can forgive him?'"
And 'Amr ibn 'Absah رضي الله عنه reported that the Prophet صلى الله عليه وسلم said: "The closest any worshipper can be to his Lord is during the last part of the night, so if you can be amongst those who remember Allaah at that time, then do so."
This is a huge incentive, then, for any person who truly desires to come closer to Allaah, all Glory and Praise be to Him, and wishes to have his sins forgiven, and his du'aa' responded to, to wake up when everyone else is asleep, and remember Allaah سبحانه و تعالى privately, and all the while praying to Him.
Furthermore, Abu Sa'eed رضي الله عنه and Abu Hurayrah رضي الله عنه both reported that the Prophet صلى الله عليه وسلم said: "Allaah waits until the last third of the night has passed, then He descends down to the skies of this world and says: 'Is there any that seeks forgiveness? Is there any who wishes to repent? Is there any who is asking? Is there any who is making a du'aa'?' And this continues, until the break of dawn."
In another wording: "Our Lord - all Glory and Praise be to Him - comes down in the last third of the night to the skies of this world, and He says: 'Who is there making a du'aa', so that I can respond to him? Who is there that is asking Me, so that I can give him? Who is there that is seeking My forgiveness, so that I can forgive him?'"
Hence why Imaam Abu Bakr at-Tartooshi رحمه الله said, "And this chapter is closed by stating: He is not a wise man who has a need to Allaah, yet he sleeps away his need at night!" 
So if a person wishes that his du'aa' be answered, this is the best time to ask.
In addition to the ahaadeeth pertaining to du'aa' at the last third of the night, there are also ahaadeeth informing us that there is a general time at night during which du'aa's are responded to. The Prophet صلى الله عليه وسلم said:
"There is at night an hour, no Muslim happens to be asking Allaah any matter of this world or the Hereafter, except that he will be given it, and this (occurs) every night."
  Notes:
 Shaykh Muhammad Mustafa al-Jibaly حفظه الله, The Dreamer's Handbook, p. 72.
 Reported by al-Bukhaari and Muslim, from Abu Hurayrah رضي الله عنه, as reported in Saheeh al-Jaami' #8021.
 Authentic, narrated by at-Tirmidhi, an-Nasaa'i, and al-Haakim from 'Amr ibn 'Absah رضي الله عنه, and authenticated by Shaykh al-Albaani رحمه الله in Saheeh al-Jaami' #1173.
 Authentic, narrated by Ahmad and Muslim from Abu Hurayrah رضي الله عنه and Abu Sa'eed al-Khudri رضي الله عنه.
 Reported by al-Bukhaari (#1145), Muslim (#758) and others.
 Shaykh Saleem al-Hilaali حفظه الله, an-Nubadh al-Mustataabah fid-Da'waat al-Mustajaabah, Daar Ibn al-Jawzi, Dammaam, 1413 A.H., p. 47.
 Reported by Muslim (#757).
 Shaykh Dr. Abu Ammar Yasir Qadhi حفظه الله, Du'a - The Weapon of the Believer, pp. 117-119.