THE BEST TIME FOR ASKING ALLAH

Middle of the night:

The Messenger (Peace and Blessing be upon him) said: "The Gates of Heaven are open at midnight, and a caller calls, 'Is there a supplicant that his supplication may be granted? Is there a petitioner that his petition maybe granted? Is there a distressed person so that his distress may be removed?' At that time, no Muslim makes a Dua but that it is answered, except for an adulteress who trades with her body, or a person who gathers his money unlawfully." [Sahih al-Jami' 2968]

At the time of Azan and between it and the Iqamah:

The Messenger (Peace and Blessings Be upon Him) said: "A Dua between Azan and the Iqamat is not refused, so make du'a at that time." [Ibn Khuzaymah, Ibn Hibban]

The Messenger (Peace and Blessing Be upon Him) said: "Two kinds of du'a are not rejected - or rarely are they rejected - du'a at the time of the call for prayer, and du'a at the time of fighting when the warriors are engaged with each other." [Abu Dawud]

On Friday after 'Asr The Messenger (Peace and Blessings Be upon Him) said: "Friday has twelve periods. Among them is a period when no Muslim asks Allah for anything but He grants it to him. Search for that period late after 'Asr." [Sahihal-Jami' 8402]

When it is raining:

The Messenger (Peace and Blessings Be upon Him) said: "Two kinds of du'a are not rejected: du'a at the time of the call for prayer, and du'a during rain." [Abu Dawud]

Being a parent, while fasting and while travelling:

The Messenger (sallallaahu 'alaihi wa sallam) said: "Three types of du'a are undoubtedly granted: a parent's du'a for his children, du'a of a fasting person, and du'a of the traveller." [Abu Dawud]

While Making Sujood:

The Prophet (Peace and Blessings Be upon Him) said: "The servant is closest to his Lord while in sujud, so ask Allah in that time."

Laylatul-Qadr: 

Aisha asked the Prophet (Peace and Blessings Be upon Him) What should I say on Laylatil-qadr? And he replied, O Allah, indeed you are a Pardoner,and you love pardon; so pardon me "Allahuma inaka afuuin tuhibul afua Fahafu Anni" .