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).
These times are as follows:

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. Late at night
When people are sleeping and busy with worldly pleasures Allah (Subhanahu wa Ta’ala) gives the believers an opportunity, or an answer hour if they can fight sleep and invoke Allah (Subhanahu wa Ta’ala) for whatever they need.

The Prophet (salAllahu alayhi wasalam) said: ‘There is at night an hour, no Muslim happens to be asking Allah any matter of this world or the Hereafter, except that he will be given it, and this (occurs) every night.’
[Muslim #757]

3. Between Adhan and Iqamah
Anas (radiAllahu anhu) narrated that Allah’s Messenger (salAllahu alayhi wasalam) said: ‘A supplication made between the Adhan and Iqama is not rejected.’
[Ahmad, abu Dawud #521, at-Tirmidhi #212, Sahih al-Jami #3408, an-Nasai, and Ibn Hibban graded it sahih (sound)]

4. 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).
(Note: when we say an hour here we do not mean the hour everyone knows (60 minutes) but an unspecified period of time because the time counters which we use today came after the Prophet (salAllahu alayhi wasalam) had died.

5. While Drinking Zamzam Water
Jaber (radiAllahu anhu) narrated that Allah’s Messenger (salAllahu alayhi wasalam) said: ‘Zamzam water is for what it is drunk for.’
[Ahmad 3: 357 and Ibn Majah #3062]

This means that when you drink Zamzam water you may ask Allah (Subhanahu wa Ta’ala) for anything you like to gain or benefit from this water such as healing from illness…… etc.

6. While Prostrating
Abu Hurairah (radiAllahu anhu) narrated that Allah’s Messenger (salAllahu alayhi wasalam), said:
‘The nearest a slave can be to his Lord is when he is prostrating, so invoke (supplicate) Allah (Subhanahu wa Ta’ala) much in it.
[Muslim, abu Dawud, an-Nasa’i and others, Sahih al-Jami #1175]

When a Muslim is in his Salat (prayer) he is facing Allah (Subhanahu wa Ta’ala) and when he prostrates he is the nearest he can be to Allah (Subhanahu wa Ta’ala) so it is best to invoke Allah (Subhanahu wa Ta’ala) at this time.
It is said that while in prostration, one should not ask for worldly needs (ie a nice car, a new job, etc), but for the Hereafter.

7. When Waking Up at Night
Narrated Ubada Bin As-Samit that Allah’s Messenger (salAllahu alayhi wasalam) said: ‘Whoever wakes up at night and says
La ilaha illallahu wahdahu la shrika lahu lahulmulku, wa lahul hamdu, wa huwa ala kulli shai’in qadir. Alhamdu lillahi, wa subhanallahi wa la ilaha illallahu, wallah akbir, wa la hawla wala quwata illa billah

(none has the right to be worshipped but Allah (Subhanahu wa Ta’ala) He is the only one who has no partners. His is the kingdom and all the praises are for Allah (Subhanahu wa Ta’ala) All the glories are for Allah (Subhanahu wa Ta’ala) And none has the right to be worshipped but Allah (Subhanahu wa Ta’ala) and Allah (Subhanahu wa Ta’ala) is the greatest and there is neither might nor power except with Allah (Subhanahu wa Ta’ala) and then says, Allahumma ighfir li (O Allah! Forgive me) or invokes Allah (Subhanahu wa Ta’ala),

he will be responded to and if he makes ablution and performs Salat (prayer), his Salat (prayer) will be accepted.
[Sahih al-Bukhari]

8. At The End Of The Obligatory Salat:
Narrated Abu Omamah (radiAllahu anhu): that Allah’s Messenger (salAllahu alayhi wasalam) was asked, O Messenger of Allah, which supplication is heard (by Allah (Subhanahu wa Ta’ala), he said the end of the night and at the end of the obligatory Salat (prayer)
[at-Tirmidhi]

This time is after saying ‘At-tahyat’, and before making Tasleem (finishing prayer)

9. The Night Of ‘Qadr’ (Decree)
This night is the greatest night of the year. This is the night which the almighty Allah (Subhanahu wa Ta’ala) said about it, “The night of Al-Qadar (Decree) is better than a thousand months.”
[Surah al-Qadr, 97: 3]

The Night of Decree is one of the odd nights of the last ten nights of the blessed month of Ramadan. The angels descend down to the earth, and the earth is overwhelmed with peace and serenity until the break of dawn when the doors of Paradise are opened, the worshipper is encouraged to turn to Allah to ask for his needs for this world and the Hereafter.