Best Times to make Du'a
There are certain times dua (supplication) is more likely to be accepted by Allah (Subhanahu wa Ta'ala) as mentioned by Prophet (salAllahu alayhi wasalam). Some Selected are mentioned below, while there are other a lot of occasions.
1- The Last Third Of The Night
Abu Hurairah (radiAllahu anhu) narrated that Allah's Messenger (salAllahu alayhi wasalam) said: 'In the last third of every night our Rabb (Cherisher and Sustainer) (Allah (Subhanahu wa Ta'ala)) descends to the lowermost heaven and says; "Who is calling Me, so that I may answer him? Who is asking Me so that may I grant him? Who is seeking forgiveness from Me so that I may forgive him?."'
[Sahih al-Bukhari, Hadith Qudsi]
Amr ibn Absah narrated that the Prophet 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 Allah at that time, then do so.'
[at-Tirmidhi, an-Nasa'i, al-Hakim - Sahih]
2- An Hour On Friday
Narrated Abu Hurairah (radiAllahu anhu): Allah's Messenger (salAllahu alayhi wasalam) talked about Friday and said:
'There is an hour on Friday and if a Muslim gets it while offering Salat (prayer) and asks something from Allah (Subhanahu wa Ta'ala), then Allah (Subhanahu wa Ta'ala) will definitely meet his demand.' And he (the Prophet (salAllahu alayhi wasalam) pointed out the shortness of that particular time with his hands.
[Sahih al-Bukhari]
Some have said that this hour is from the time the Imam (prayer's leader) enters the mosque on Friday's prayer until the prayer is over (ie between the two khutbahs), whereas others have said that it is the last hour of the day (ie after the Asr prayer until the Maghrib prayer).
3- Dua of the one fasting until he breaks his fast
The Messenger of Allah (salAllahu alayhi wasalam) said; Three supplications will not be rejected (by Allah (Subhanahu wa Ta'ala)), the supplication of the parent for his child, the supplication of the one who is fasting, and the supplication of the traveler.
[al-Bayhaqi, at-Tirmidhi - Sahih]
4- Dua of the one fasting at the time of breaking fast
The prophet (salAllahu alayhi wasalam) declared,
'Three men whose dua is never rejected (by Allah) are: when a fasting person breaks fast (in another narration, the fasting person until he breaks his fast), the just ruler and the one who is oppressed.'
[Ahmad, at-Tirmidhi - Hasan]
When Muslims gather for the purpose of invoking and remembering Allah (Zikrullah)
The Prophet (salAllahu alayhi wasalam) said:
If a group of people sit together remembering Allah, the angels will circle them, mercy will shroud them, peace will descend onto them and Allah will remember them among those with Him.
[Muslim]