Why Doesn’t Allah Answer Your Dua?
Question

Why doesn't Allah answer our Du`a?
Praise be to Allah.The power of Du`a
Imam Ibn Al-Qayyim (may Allah have mercy on him) said:
“Du`a and ta`awwudhat [prayers seeking refuge with Allah] are like a weapon, and a weapon is only as good as the person who is using it; it is not merely the matter of how sharp it is. If the weapon is perfect and free of faults, and the arm of the person using it is strong, and there is nothing stopping him, then he can lay waste the enemy. But if any of these three features is lacking, then the effect will be lacking accordingly.” (Ad-Da’ wad-Dawa’, p. 35)
From this statement, it will be clear that there is an etiquette and rulings which must be fulfilled, in the Du`a and in the person making the Du`a. 
Conditions of dua
There are also things that may prevent the Du`a from reaching Allah or being answered . These things must be removed from the person making the Du`a and from the Du`a. When all of these conditions are fulfilled , then the Du`a may be answered.

Sincerity in making Du`a. This is the most important condition. Allah has commanded us to be sincere when making Du`a, as He says (interpretation of the meaning):

So, call you (O Muhammad and the believers) upon (or invoke) Allah making (your) worship pure for Him (Alone) (by worshipping none but Him and by doing religious deeds sincerely for Allah’s sake only and not to show off and not to set up rivals with Him in worship.) [Ghafir 40:14]
Sincerity in Du`a means having the firm belief that the One upon Whom you are calling Allah, may He be Glorified and Exalted is alone able to meet your need, and it also means avoiding any kind of showing off in your Du`a.

Repentance and turning back to Allah. Sin is one of the main reasons why Du`as are not answered, so the person who is making Du`a should hasten to repent and seek forgiveness before he makes Du`a. Allah tells us that Nuh (peace be upon him) said:

I said (to them): Ask forgiveness from your Lord, verily, He is Oft-Forgiving; He will send rain to you in abundance, And give you increase in wealth and children, and bestow on you gardens and bestow on you rivers. [Nuh 71:10-12]

Beseeching, humbling oneself, hoping for Allah's reward and fearing His punishment. This is the spirit, essence and purpose of Du`a. Allah says (interpretation of the meaning): Invoke your Lord with humility and in secret. He likes not the aggressors [Al-`Araf 7:55]
Urgently beseeching and repeating the Du`a, without getting exasperated or bored; this is achieved by repeating the Du`a two or three times. Restricting it to three times is preferable, in accordance with the Sunnah of the Prophet (peace and blessings of Allah be upon him). Ibn Mas`ud (may Allah be pleased with him) narrated that the Prophet (peace and blessings of Allah be upon him) liked to say a Du`a three times and ask for forgiveness three times. (Narrated by Abu Dawud and An-Nasa’i)
Making Du`a at times of ease, and saying more Du`as at times of plenty. The Prophet (peace and blessings of Allah be upon him) said: Remember Allah during times of ease and He will remember you during times of hardship. (Narrated by Ahmad)
Seeking to draw closer to Allah by calling upon Him by His Most Beautiful Names and Sublime Attributes at the beginning of the Du`a or at the end. Allah says (interpretation of the meaning): And (all) the Most Beautiful Names belong to Allah, so call on Him by them [Al-A'raf 7:180].
Choosing the clearest and most concise words and the best of Du`as. The best of Du`as are the Du`as of the Prophet (peace and blessings of Allah be upon him), but it is permissible to say other words according to the specific needs of a person.

Other aspects of the etiquette of making Du`a, although they are not obligatory, are: to face the Qiblah; to make Du`a in a state of purity; to start the Du`a with praise of Allah and blessings upon the Prophet (peace and blessings of Allah be upon him). It is also prescribed to raise the hands when making Du`a.
One of the things that helps to bring a response to Du`a is to seek the best times and places .
Among the best times is the time just before Fajr (dawn), the last third of the night , the last hour of Jumu`ah (Friday), when rain is falling , and between the Adhan and Iqamah .
Among the best places are mosques in general, and Al-Masjid Al-Haram [in Makkah] in particular.
Among the situations in which Du`a is more likely to be answered are: when one is mistreated or oppressed, when one is travelling, when one is fasting, when one is in desperate need, and when a Muslim makes Du`a for his brother in his absence.
Why is your Du`a not answered?
Things which may prevent Du`a from being answered include:

When the Du`a is weak in itself, because it involves something inappropriate, or involves bad manners towards Allah, may He be Exalted, or it is inappropriate, which means asking Allah for something which it is not permitted to ask, e.g. when a man asks to live forever in this world, or he asks for a sin or something prohibited, or he prays that he will die, and so on. 

Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: A person's Du`as will continue to be answered so long as he does not pray for something sinful or for the breaking of family ties. (Narrated by Muslim)

When the person who is making Du`a is weak in himself, because he is faint-hearted in his turning towards Allah. This may be either because of bad manners towards Allah, may He be Exalted such as raising his voice in Du`a or making Du`a in the manner of one who thinks he has no need of Allah; or because he pays too much attention to the wording and tries to come up with unnecessarily ornate phrases, without paying attention to the meaning; or because he tries too hard to weep or shout without really feeling it, or he goes to extremes in that.
The reason why his Du`a is not answered may be because he has done something that Allah has forbidden , such as having prohibited wealth whether it be food or drink or clothing or accommodation or transportation, or he has taken a prohibited job, or the stain of sin is still in his heart, or he is following Bid`ah (innovation) in religion, or his heart has been overtaken by negligence.
Consuming prohibited wealth is one of the major reasons why Du`as are not answered.

Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: O people, Allah is Good and only accepts that which is good. Allah commanded the pious to follow the same commandments as He gave to the Messengers. He says (interpretation of the meaning):
“O (you) Messengers! Eat of the Tayyibat [all kinds of lawful foods which Allah has made lawful] and do righteous deeds. Verily, I am Well-Acquainted with what you do.” [Al-Mu’minun 23:51]
“O you who believe (in the Oneness of Allah Islamic Monotheism)! Eat of the lawful things that We have provided you with.” [Al-Baqarah 2:172]
Then he (peace and blessings of Allah be upon him) mentioned a man who has travelled on a long journey and is dishevelled and covered with dust; he stretches forth his hands to the heaven, (saying) O Lord, O Lord, but his food is prohibited, his drink is prohibited, all his nourishment is prohibited, so how can his Du`a be accepted? (Narrated by Muslim)
The man described by the Prophet (peace and blessings of Allah be upon him) had some of the characteristics which help Du`as to be answered, he was travelling and he was in need of Allah, may He be Exalted and Glorified but the fact that he consumed prohibited wealth prevented his Du`a from being answered. We ask Allah to keep us safe and sound.

Trying to hasten the response. Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: The Du`a of any one of you will be answered so long as he is not impatient and says, I made Du`a but it was not answered. (Narrated by Al-Bukhari and Muslim)
Making the Du`a conditional, such as saying, O Allah, forgive me if You will or O Allah, have mercy upon me if You will. The person who makes Du`a has to be resolute in his supplication, striving hard and earnestly repeating his Du`a. The Prophet (peace and blessings of Allah be upon him) said: Let not any one of you say, O Allah, forgive me if You will, O Allah, have mercy on me if You will. Let him be resolute in the matter, whilst knowing that no one can compel Allah to do anything. (Narrated by Al-Bukhari and Muslim)

In order for Du`as to be answered, it is not essential to adhere to all of these points and be free of all that could prevent one's Du`a from being answered. That is something which happens very rarely. But one has to try hard and strive towards achieving this.
How does Allah answer our Du`a?
Another important point is to realise that the response to the Du`a may take different forms:

either Allah will respond and fulfil the desire of the person who made the Du`a, or He will ward off some evil from him because of the Du`a, 
or He will facilitate for him something better than what he asked for, 
or He will save it with Him for him on the Day of Resurrection when he will be most in need of it.

And Allah knows best.