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). 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 worldy 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: 'Whomever 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 most Great 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 and when he doors of Paradise are opened, the worshipper is encouraged to turn to Allah to ask for his needs for this world and the Hereafter.

10. During The Rain

Narrated Sahel Ibn Sa'ad (radiAllahu anhu): that the Messenger of Allah (salAllahu alayhi wasalam) said: 'Two will not be rejected, Supplication when the Adhan (call of prayer) is being called, and at the time of the rain'. [Al-Hakim 2: 114, and Abu Dawud #2540, ibn Majah] 'Seek the response to your du'as when the armies meet, and the prayer is called, and when rain falls'[reported by Imam al-Shafi' in al-Umm, al-Sahihah #1469] The time of the rain is a time of mercy from Allah (Subhanahu wa Ta'ala) so, one should take advantage of this time when Allah (Subhanahu wa Ta'ala) is having mercy on His slaves.

11. At The Adhan

'Seek the response to your du'as when the armies meet, and the prayer is called, and when rain falls'[reported by Imam al-Shafi' in al-Umm, al-Sahihah #1469] In another hadith; 'When the prayer is called, the doors of the skies are opened, and the du'a is answered'[al- Tayalisi in his Musnad #2106, al-Sahihah #1413]

12. The One Who Is Suffering Injustice And Opression

The Messenger of Allah (salAllahu alayhi wasalam) said to Mua'ad Ibn Jabal (radiAllahu anhu), 'Beware of the supplication of the unjustly treated, because there is no shelter or veil between it (the supplication of the one who is suffering injustice) and Allah (Subhanahu wa Ta'ala)' [Sahih Al-Bukhari and Muslim] The prophet (salAllahu alayhi wasalam) declared,'Three men whose dua is never rejected (by Allah) are: the fasting person until he breaks his fast (in another narration, when he breaks fast), the just ruler and the one who is oppressed.'[ Ahmad, at-Tirmidhi - Hasan] In another hadith; The Prophet (salAllahu alayhi wasalam) declared:'Three du'as are surely answered: The du'a of the oppressed, the du'a of the traveler, and the du'a of the father/mother (upon their child)'The One who is suffering injustice is heard by Allah (Subhanahu wa Ta'ala) when he invokes Allah (Subhanahu wa Ta'ala) to retain his rights from the unjust one or oppressor. Allah (Subhanahu wa Ta'ala) has sworn to help the one who is suffering from injustice sooner or later as the Messenger of Allah (salAllahu alayhi wasalam) said.

13. The Traveler

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] During travel supplication is heard by Allah (Subhanahu wa Ta'ala) if the trip is for a good reason, but if the trip is for a bad intention or to perform illegal things (making sins) this will not apply to it.

14. The Parent's Supplication For Their Child

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]

15. Dua After Praising Allah And Giving Salat On The Prophet (Salallahu Alayhi Wasalam) in the tashahhud at the end of salat.
Narrated Faddalah ibn Ubayd (radiAllahu anhu): that the Messenger of Allah (salAllahu alayhi wasalam) said: 'When anyone of you makes du'a, let him start by glorifying his Lord and praising Him, then let him send blessings upon the Prophet (salAllahu alayhi wasalam), then let him pray for whatever he wants'[abu Dawud #1481, at-Tirmidhi #3477]In another hadith; Baqiy ibn Mukhallid (radiAllahu anhu) narrated that that the Messenger of Allah (salAllahu alayhi wasalam) said: 'Every du'a is not responded to until one sends blessings upon the Prophet (salAllahu alayhi wasalam)'[al- Bayhaqi]In another hadith; Umar (radiAllahu anhu) narrated that the Prophet (salAllahu alayhi wasalam) said:'Du'a is detained between the heavens and the earth and no part of it is taken up until you send blessings upon your Prophet (salAllahu alayhi wasalam)'[at- Tirmidhi #486] After a person has finished his tashahhud and before saying the 'salam', supplication at this time is one likely to be responded to.Ibn Mas'ud narrates: I was once praying, and the Prophet (salAllahu alayhi wasalam), Abu Bakr and Umar (were all present). When I sat down (in the final tashahhud), I praised Allah, then sent salams on the Prophet, then started praying for myself. At this, the Prophet (salAllahu alayhi wasalam) said:'Ask, and you shall be given it! Ask, and you shall be given it!'[at-Tirmidhi #593 - hasan, Mishkat al-Misbah #931]

16. The dua of a Muslim for his absent brother or sister Muslim stemming from the heart.

The prophet (salAllahu alayhi wasalam) said:'There is no believing servant who supplicates for his brother in his absence where the angels do not say, 'the same be for you''[Muslim]

17. Dua on the Day of Arafat

The Messenger of Allah (salAllahu alayhi wasalam) said: 'The best supplication is the supplication on the day of Arafat'. [at-Tirmidhi and Malik] The day of Arafat is the essence and pinnacle of Hajj. On this great and momentous day, millions of worshippers gather together on one plain, from every corner of the world, with only one purpose in mind - to respone to the call of their Creator. During this auspicious day, Allah does not refuse the requests of His worshippers.

18. Dua during the month of Ramadan

Ramadan is month full of many blessings, thus the du'a of Ramadan is a blessed one. This can be inferred from the Prophet (salAllahu alayhi wasalam) saying:'When Ramadan comes, the Doors of Mercy (another narration says Paradise) are opened, and the doors of Hell are closed, and the Sahitans are locked up'Thus, it is clear that du'a during Ramadan has a greater chance of being accepted, as the Gates of Paradise and Mercy are opened.[Sahih al-Bukhari #1899, Muslim #1079 and others]

19. Dua when the Armies meet

When the Muslim is facing the enemy in battle, at this critical period, the du'a of a worshipper is accepted.Sahl ibn Sa'd (radiAllahu anhu) narrtaed the the Prophet (salAllahu alayhi wasalam) said:'Two duas are never rejected, or rarely rejected: the du'a during the call for prayer, and the du'a during the clamity when the two armies attack each other'[abu Dawud #2540, ibn Majah, al-Hakim]In another narration: 'Seek the response to your du'as when the armies meet, and the prayer is called, and when rain falls'[reported by Imam al-Shafi' in al-Umm, al-Sahihah #1469]

20. When Muslims gather for the purpose of invoking and remembering Allah (Dhikrullah)

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]

21. First Ten days of Dhul-Hijjah

The Prophet (salAllahu alayhi wasalam) said:'There are no days during which good deeds are more beloved to Allah than during these ten days'[Sahih al-Bukhai #969 and others]

22. Dua when the heart reaches out to Allah and is ready to be totally sincere

23. At Midnight

Abu Umamah (radiAllahu anhu) said, the Prophet (salAllahu alayhi wasalam) was questioned; 'Which du'a is heard (by Allah)?' He answered,'At midnight and at the end of every obligatory prayer.'[at- Tirmidhi - Hasan]

25. Dua of people after the death of a person

In a long hadith, Umm Salamah (radiAllahu anha) narrated that the Prophet (salAllahu alayhi wasalam) said, when Abu Salamah had just passed away, and had closed his eyes,'Do not ask for yourselves anything but good, for the angels will say 'Ameen' to all that you ask for. O Allah, forgive Abu Salamah, and raise his ranks among those who are guided.'[Muslim, abu Dawud, Ahmad]

26. 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]

27. 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]

29. Dua of a just Ruler

The prophet (salAllahu alayhi wasalam) declared,'Three men whose dua is never rejected (by Allah) are: the fasting person until he breaks his fast (in another narration, when he breaks fast), the just ruler and the one who is oppressed.'[ Ahmad, at-Tirmidhi - Hasan]

Surah Isra 17 verse 80 Say: "O my Lord! let my entry be by the Gate of Truth and Honor and likewise my exit by the Gate of Truth and Honor; and grant me from Thy Presence an authority to aid (me)."

Ameen,Thumameen.
Transliteration :Wa qur rabbi adkhilni mudkhala sidqiw wa akhrijni mukhraja sidqiw wa-j'al li mil ladunka sulta_nan nasira_(n).

Al-Tirmidhi HadithHadith 2482 Narrated by AbuHurayrah (May Allah be pleased with him)
Allah's Messenger (peace be upon him) used to say, " O Allah, grant me benefit in what Thou hast taught me, teach me what will benefit me, and increase my knowledge. Praise be to Allah in all circumstances. I seek refuge in Allah from the state of those who go to Hell."
Tirmidhi and Ibn Majah transmitted it, Tirmidhi saying this is a tradition whose isnad is gharib.

Transliteration:an Abi Hurairah (radiya Allahu anhu) qaal:
qaala Rasul Allah (salla Allahu alayhi wa sallam): "Allahumma infa'ni bima 'allamtani, wa allimni maa yanfa'oni wa zidni 'ilman, alhamdulillahi alaa kolli haal, wa a'odtho billahi min haali ahlil naar."