During this blessed night,  when the angels descend down to earth with the angel Jibreel عليه السلام, and when the earth is overwhelmed with peace and serenity until the break of dawn, and when the doors of Paradise are opened, the worshipper is encouraged to turn to Allaah سبحانه و تعالى, and to ask of his needs for this world and the Hereafter.
As Allaah سبحانه و تعالى says, "Verily! We have sent it (this Qur'aan) down in the night of Al-Qadr (Decree). And what will make you know what the night of Al-Qadr (Decree) is? The night of Al-Qadr (Decree) is better than a thousand months (i.e. worshipping Allaah in that night is better than worshipping Him a thousand months, (i.e. 83 years and 4 months)). Therein descend the angels and the Rooh [Jibraa'eel (Gabriel)] by Allaah's Permission with all Decrees. Peace! (All that night, there is Peace and Goodness from Allaah to His believing slaves) until the appearance of dawn."
Ummul Mu'mineen 'Aa'ishah رضي الله عنها narrated that she asked the Prophet صلى الله عليه وسلم: "O Messenger of Allaah! What du'aa' should I make on Laylatul-Qadr (the Night of Decree)?" He صلى الله عليه وسلم answered:
قولي: اللَّهُمَّ إِنَّكَ عَفُوٌّ تُحِبُّ العَفْوَ فَاعْفُ عَنِّي
Allaahumma innaka 'afuwwun tuhibbul-'afwa fa'fu 'annee
"Say, 'O Allaah! You are Forgiving, and love to forgive, so forgive me!'"
  Notes:
 The Night of Decree is one of the odd nights of the last ten nights of Ramadaan.
 Narrated by at-Tirmidhi, Ibn Maajah, and al-Haakim and authenticated in Saheeh al-Jaami' # 4423.
 Shaykh Dr. Abu Ammar Yasir Qadhi حفظه الله, Du'a - The Weapon of the Believer, pp. 128-129.