There is a special hour on Friday (Jumu'ah), during which all prayers are accepted and answered by Allaah, all Glory and Praise be to Him. This is part of the blessings that Allaah سبحانه و تعالى has blessed this day with, over all the other days of the week. 

Abu Hurayrah رضي الله عنه narrated that the Prophet صلى الله عليه وسلم said: 

فِيهِ سَاعَةٌ لاَ يُوَافِقُهَا عَبْدٌ مُسْلِمٌ، وَهْوَ قَائِمٌ يُصَلِّي، يَسْأَلُ اللَّهَ تَعَالَى شَيْئًا إِلاَّ أَعْطَاهُ إِيَّاهُ ‏.‏ وَأَشَارَ بِيَدِهِ يُقَلِّلُهَا‏ 

"There is an hour (on Friday) during which, if any Muslim is standing in front of Allaah تعالى in salaah, and asking Allaah تعالى for something good, he will be granted his request." Then he made a sign with his hands, showing that it was a very short period of time. [1] 

There are numerous opinions as to the exact hour of Friday during which this occurs, but two opinions are the strongest: 

1) When the Imaam sits down between the two khutbahs until the end of the khutbah, 

2) After the 'Asr prayer until the Maghrib prayer. 

Imaam Ibn al-Qayyim رحمه الله preferred the second over the first one. [2] 

This is because of the following hadeeth: 

يَوْمُ الْجُمُعَةِ ثِنْتَا عَشْرَةَ ‏"‏ ‏.‏ يُرِيدُ سَاعَةً ‏"‏ لاَ يُوجَدُ مُسْلِمٌ يَسْأَلُ اللَّهَ عَزَّ وَجَلَّ شَيْئًا إِلاَّ آتَاهُ اللَّهُ عَزَّ وَجَلَّ فَالْتَمِسُوهَا آخِرَ سَاعَةٍ بَعْدَ الْعَصْرِ 

"Friday has twelve hours (or parts to it). There is one hour during which if a Muslim asks Allaah عز و جل anything, Allaah عز و جل will give it to him, so find it during the last hour after 'Asr." [3] 

This is also the opinion of the vast majority of early scholars رحمة الله عليهم as also of the Prophet's صلى الله عليه وسلم Companions رضي الله عنهم أجمعين - that this hour occurs after 'Asr prayer, right before sunset. [4] [5] 

Notes: 

[1] Narrated by al-Bukhaari (935) from Abu Hurayrah رضي الله عنه. 

[2] Zaad al-Ma'aad, 1/378-396 

[3] Reported by Abu Dawood (#926 of Saheeh as-Sunan) and an-Nasaa'i (#1316 of Saheeh as-Sunnah). 

[4] See Shaykh Saleem al-Hilaali's حفظه الله discussion of this point in an-Nubadh al-Mustataabah fid-Da'waat al-Mustajaabah, Daar Ibn al-Jawzi, Dammaam, 1413 A.H., pp. 50-55. 

[5] Shaykh Dr. Abu Ammar Yasir Qadhi حفظه الله, Du'a - The Weapon of the Believer, p. 126.

See also: 

Dua (Supplication)

Jumu'ah (Friday)