The word “Dua” in Arabic means "calling", it is the act of remembering Allah and calling upon Him. Dua, Supplication is a powerful connection between Allah and You spiritually. Supplication signifies Allah’s ability and our inability, which alleviates worry, removes distress, and eases a person’s affairs. Allah SWT has allowed believers to ask from Him for whatever they need and Has promised to answer them. 

It is an act of worship:  Dua/Prayer is an act of obedience to Allah’s command and a wordy expression of our submission and piety to Him. Allah SWT says, "And your Lord says: "Call on Me; I will answer your (Prayer), but those who are too arrogant to serve Me will surely find themselves in Hell - in humiliation" [40: 60].  -  Prophet PBUH said, “Supplication is the essence of worship” [Tirmidhi].  -  Allah SWT says, "When My servants ask about Me, I am indeed close to them.  I listen to the prayer of every supplicant, when he calls on Me.  Let them also, with a will, listen to My call, and believe in Me, so that they may walk in the right way" [2:186].

Etiquette of Dua:  Allah knows all of our secrets, dreams, and wishes, nothing is hidden from Him. Yet He wants us to ask Him for what we need.  Prophet PBUH said, “Supplicate Allah with confidence, He will answer you” [Tirmidhi].  -  Every Dua should begin with the praise of Allah, the recognition that He alone has the Power and Might over everything. When supplicating, heart should be filled with humility, a soft heart and tears of remorse are the best weapons of a human being in his\her quest to achieve nearness to Allah SWT. When one prays emotionally and the Dua is sincere then its acceptance is assured. When making Dua, have confidence and trust that Allah will answer your supplications. 

Virtues of Dua:  Dua pleases the Almighty and brings believers closer to Him.

Why some Supplication remains unanswered? Allah does not look at our forms, shapes or gender; He looks into sincerity of our hearts. If one’s heart is filled with arrogance and making dua is just a statement of the tongue then how can we expect its acceptance? Prophet PBUH said, “Allah does not accept the supplication of an unguarded heart” [Tirmidhi].  Whoever asks something from Allah, his dua is always heard and answered. Sometimes it is not answered in the way the supplicant expects, Allah may respond to it in way that is better for him\her. Or, He SWT might keep some evil away from the supplicant; Or  He SWT will reward him\her in the hereafter. It is important to remember that Allah knows what is best for us. Allah says, “... it may be that you dislike a thing which is good for you and that you like a thing which is bad for you. Allah knows but you do not know” [2: 216].  -  Prophet PBUH said, “A person’s duas will be answered so long as he does not pray for something sinful or for the breaking of family ties.” [Muslim]  -  Prophet PBUH said, "The dua of any one of you will be answered so long as he is not impatient and says, ‘I made dua but it was not answered" [Bukhari].  Prophet PBUH said, “The invocation of anyone of you is responded to by Allah SWT if he does not show impatience by saying; 'I invoked Allah but my request has not been granted.’” [Bukhari]

Making Dua for others:  Whoever prays for others it pleases the Almighty greatly.  Nothing is more swiftly answered than a Dua recited for others.

Times of Supplication:  We can ask Allah at any time of the day or night for whatever our hearts desire and know that He will hear us and respond to our calls.  There are certain times supplication is more likely to be accepted by Allah SWT as it has been narrated in some hadiths, -  

The last third of the night:  [Bukhari]. -   

An Hour on Friday [Bukhari].  -  

At the end of Obligatory(Fard) Salat:   [Tirmidhi].  -  

On the Day of Arafa(Hajj) : [Tirmidhi].