Situations in which Dua is more likely to be answered
When one is mistreated or oppressed
When one is travelling
The Dua of a Parent against their child
This Hadith refers to the above situations: "Three kinds of Dua are surely answered; the Dua of an oppressed, the Dua of a traveller and the Dua of a parent against his son or daughter." ( Abu-Dawud and At-Tirmidhi)
When one is fasting
Narrated Abu Hurayrah (Radhi Allahu Anhu) the Prophet (Sallallahu 'Alayhi wa Sallam) said: “Three persons will not have their supplication rejected: the just ruler, the fasting person until breaking the fast and the supplication of the oppressed person.” (At-Tirmidhi, ibn Majah)
When one is in desperate need
“Is not He (better than your gods) Who responds to the distressed one, when he calls Him, and Who removes the evil, and makes you inheritors of the earth, generations after generations? Is there any God with Allah? Little is that you remember! (Surah An-Naml 27: 62)
We must remember that Allah Subhanahu wa Ta'ala answers the supplication of Muslims or non-Muslims if a) The invocation of the one in dire need even if the person is a disbeliever as long as there is a real need – ill, poor, afflicted by calamity and b) The invocation of one who is oppressed or injustice done to him.
When a Muslim makes Dua for his brother in his absence.
Abu Dharr (Radhi Allahu Anhu) reported that Allah's Messenger (Sallallahu Alayhi wa Sallam) said: “There is no believing servant who supplicates for his brother behind his back (in his absence) that the Angels do not say: The same be for you too.” (Muslim)
The Prophet (Sallallahu 'Alayhi wa Sallam) said: “The supplication that gets quickest answer is the one made by the Muslim for another in this absence. (Abu Dawud, At-Tirmidhi)
Gatherings in which Allah is remembered
It was narrated by Abu Hurairah (Radhi Allahu Anhu) in a long Hadith found both in Sahih Al-Bukhari and Muslim that the Prophet (Sallallahu 'Alayhi wa Sallam) said that Allah has some angels who are on the lookout for those who remember Him in a gathering and encircle them.