The word du'aa is the plural of da'wah and both come from da'aa which means to ask or to call.  To call "to something" in this way is to encourage others to do it.  The meaning of du'aa in Islam is: for the slave to call out to his Lord to care for him, have mercy on him and supply him with his needs and wants. 
The reality of du'aa is to manifest one's absolute poverty and powerlessness before Allah Most High and the affirmation that all power to bring about anything belongs exclusively to Allah Most High.  It is a sign of true worship and it contains the meaning of praising and exalting Allah Most High and ascribing to Him the attributes of generosity and grace.

Types of Du'aa Mentioned in the Qur'an

1. Worship.  Asking for something which only Allah can grant is an act of worship which can only be directed to Allah Most High.  To address it to other than Allah is shirk.
   
   {Wa laa tad'u min duni Allahi ma laa yanfa'uka wa laa yadhurruka}
   {And do not call besides Allah those who can not benefit your and cannot harm you.} Yunus:106
   
   
2. Seeking of Aid.  Calling to one for aid can only be directed to Allah in issues where only He can respond.  One can request aid from a present, living and capable individual and this is not shirk.  However, to ask those incapable of helping (such as the deceased) or not present is to worship other than Allah Most High.   Allah rejected the ability of anyone to help anyone else in disobedience to Him on Qiyama in the following:
   
   {Ud'oo shuhadaa'akum min duni Allahi in kuntum saadiqeen.}
   {Call your witnesses beside Allah if your are indeed truthful.} Al-Baqarah:23
   
   
3. Asking for something.  Allah said:
   
   {Ud'ooniy astajib lakum}
   {Call to (ask) Me that I may respond to you.} Ghaafir:60
   
   
4. Exalting of Allah and addressing one another.  Allah said about the people of Paradise:
   
   {Da'waahum fihaa subhaanaka Allahumma wa tahiyyatuhum fihaa salaam.}
   {Their call therein is "Exalted are You, O Allah!" and their greeting therein is "Salaam".} Yunus:10
   
   
5. Allah calling us.
   
   {Yauma yad'ookum fatastajiboona bihamdihi}
   {On the day when He calls you and you respond with His praise.}  Al-Israa:52
   
   
6. Calling to Allah in praise and supplication.
   
   {Qul id'oo Allaha au id'oo Ar-Rahman}
   {Say:  Call "Allah" or call "Ar-Rahman"...}   Al-Israa:110

Those Whose Du'aa is Answered by Allah

Allah and His Prophet (sas) have mentioned various people/conditions under which Allah answers the Du'a of His slaves.  Among them:

1. The one in desparate need.
2. The oppressed upon whom injustice is being inflicted
3. A parent who makes du'a against their child
4. A just ruler
5. A righteous individual
6. The righteous son/daughter who prays for his parents
7. The fasting individual until they break their fast
8. Any Muslim who makes du'aa for another Muslim without their knowledge
9. Every Muslim as long as they are not asking for injustice or breaking of family ties.
10. The one making tauba (repentance) from sin
11. The traveller whose sustenance is from halal

Prerequisites of Du'aa

When calling to your Lord, be sure to be aware of the following requirements for making du'aa:

1. Sincerity of directing all of your worship to Allah alone
2. Obedience to Allah by fulfilling all obligations upon you and avoiding all that forbidden.
3. Avoiding all harm consumption of food, clothing/jewelry and drink
4. Preceding the du'aa with a good deed
5. Purity (tahaara)

Some Etiquettes of Du'aa

Also, pay close attention to the following etiquettes (aadaab) of supplicating Allah Most High:

1. Take advantage of the blessed times which Allah and His Prophet (sas) have told us about such as the day of Arafat, all of Ramadhan (especially the last ten), Jumu'ah and the time of tahajjud (after sleeping and before fajr), etc.  Allah said:
   
   {Wa bil as-haari hum yastaghfirun}
   {And in the late night they seek forgiveness}  Adh-Dhaariyaat:18
   
   
2. Take advantage of the special circumstances when du'aa especially encouraged such as when one is in prostration (sajdah), upon meeting the enemy army, when Allah send His mercy (e.g., rain) and between the adhaan and the iqaama.   The Prophet(sas) said:
   
   "Aqrabu maa yakoonu al-'abdu min rabbihi 'azza wa jalla wa huwa saajid, fa akthir ad-du'aa."
   "The closest a slave ever is to his Lord is when he is in prostration, so make much supplication."  Muslim & others
   
   
3. Face the qibla.  Jaabir reports:
   
   "Anna rasulu Allahi ataa al-mawqifa bi 'arafaati wa istaqbala al-qibla wa lam yazal yad'oo hatta gharibat ash-shams."
   "The Prophet (sas) came to the place of standing at Arafat, faced the Qibla and continued making du'aa until the sun set."  Muslim
   
   
4. Lowering the voice and keeping it "in between" neither complete silence nor completely out loud.  Abu Musa Al-Ash'ariy reported that:  We were approaching with Allah's Messenger (sas).  When we came near Al-Madinah, he said "Allahu Akbar".  The people, too said "Allahu Akbar" and they raised their voices whereupon the Prophet (sas) said:
   
   "Yaa ayyuhaa an-naasi, inna alladhiy tad'oona laisa bi asammu wa laa ghaa'ibun.   Inna alladhiy tad'oona baina a'naaqi rikaabikum."
   "O people, verily the one whom you are calling is not deaf and is not absent.  Verily, the one whom you are calling is between the necks of your mounts."  Bukhari & Muslim
   
   
5. Humility, fear (of Allah and His punishment) and desire (for His mercy and His reward) when making du'aa.  Allah said:
   
   {Innahum kaanoo yusaari'oona fiy al-khairaati wa yad'oonanaa raghaban wa rahaban wa kaanoo lanaa khaashi'een.}
   {They used to race one another to good deeds and they would call to us in desire and in fear and they were humble before Us.}  Al-Anbiyaa:90
   
   
6. Be serious in what you ask for and certain of the ability and the promise of Allah to respond to your call.  Do not put conditionals on what you ask for except for things like "if it is the best for me in this life and the hereafter in Your knowledge", but not things like "if you wish", etc.  Allah does not respond to du'aa which is made "routinely" or frivolously without consciousness.  The Prophet (sas) said:
   
   "Ud'oo Allaha wa antum muqineen bil ijaabati.  Wa i'lamo anna Allaha laa yastajeebu du'aa'a min qalbin ghaafilin laahin."
   "Call to Allah with certainty of being answered.  And know that Allah does not respond to the du'a from a heart which is forgetful or frivolous."  At-Tirmidhi
   
   
7. To be persistant, even insistant in du'aa and repeat it three times.  Ibn Masood reported that: "When the Prophet (sas) made du'aa, he used to make it three times and when he asked, he asked three times."  Muslim
8. To begin du'aa with the mention of Allah and invoking His blessing on the Prophet (sas) before asking for things and to close one's du'aa with praise of Allah.
9. To repent from one sin's and to undo any injustices done to others (including returning property improperly taken or undoing any other kind of harm) before turning to Allah in du'aa.

When we respond to Allah's call to worhip Him alone and to obey Him by performing our obligations and avoiding all the forbidden, Allah responds to this by granting us our du'aa.   There can be nothing greater in this world than to have a du'aa which is answered by Allah Most High.  Any nation which possesses many people who have this will succeed in this world and in the next.  Allah will place in power over them those who fear and obey him and grant them victory over their enemies.  The du'aa of Muslims for each other is more powerful than Muslims only making du'aa for themselves.  Allah said:

{Wa idhaa sa'alaka 'ibaadi 'anniy fa inniy qareebun ujeebu da'wata ad-daa'i idhaa da'aani.  Fal yastajeeboo liy wa li yu'minoo biy la'allahum yarshudoon.}

{And when my slave asks you about me, I am surely close.  I answer the du'aa of the caller when he calls me. So, let him respond to Me and believe in Me that perchance he may be guided.}  Al-Baqarah:186

{Wa qaala rabbukum ud'ooniy astajib lakum.  Inna alladheena yastakbiroona 'an 'ibaadatiy sayadkhuloona jahannama daakhireen.}

{And your Lord said:  Call to me that I may respond to you.   Verily, those who are too arogant to worship me will enter Hell-fire humiliated.}   Ghaafir:60