Prayer - αἰτέω - Begging

αἰτέω – aiteō
1SA 2:7 The LORD sends poverty and wealth; he humbles and he exalts.
1SA 2:8 He raises the poor from the dust and lifts the needy from the ash heap;
he seats them with princes and has them inherit a throne of honor.
"For the foundations of the earth are the LORD's; upon them he has set the world.
1SA 2:9 He will guard the feet of his saints, but the wicked will be silenced in darkness.

Aiteō is the prayer of Begging. Out of an inner need, a desire or a requirement, we beg God for something. This is not a simple asking, but it is more. Jesus said, "Ask and it will be given to you; seek and you will find; knock and the door will be opened to you. For everyone who asks receives; he who seeks finds; and to him who knocks, the door will be opened. Which of you, if his son asks for bread, will give him a stone? Or if he asks for a fish, will give him a snake? If you, then, though you are evil, know how to give good gifts to your children, how much more will your Father in heaven give good gifts to those who ask him! So in everything, do to others what you would have them do to you, for this sums up the Law and the Prophets.” Mt 7:7-12

People beg for many things out of desperation. Zechariah begged for a table so he could name his son John (Lk 1:63). James and John (Mk 10:35) and their mother (Mk 10:35) begged Jesus to seat them on his right and left. The multitude begged Pilate for Barabbas (Mt 27:20); and Joseph of Aramathea begged Pilate for the body of Jesus (Mt 27:58). The warden in Philippi begged for a light to see if the prisoners escaped (Ac 16:29). Paul begged that Christians might be filled with the knowledge of his will through all spiritual wisdom and understanding. (Col. 1:9)

PS 72:12 For he will deliver the needy who cry out, the afflicted who have no one to help.
PS 72:13 He will take pity on the weak and the needy and save the needy from death.
The lame man at the Temple begged for money – and the Lord gave him legs to walk on. (Ac 3:2) God is able to do immeasurably more than all we beg for or imagine. (Eph 3:20) The Lord wants us to pray out of need just like the beggar.
Consider it pure joy, my brothers, whenever you face trials of many kinds, because you know that the testing of your faith develops perseverance. Perseverance must finish its work so that you may be mature and complete, not lacking anything. If any of you lacks wisdom, he should ask God, who gives generously to all without finding fault, and it will be given to him. But when he asks, he must believe and not doubt, because he who doubts is like a wave of the sea, blown and tossed by the wind. (James 1:2-6)
JAS 4:2 You want something but don't get it. You kill and covet, but you cannot have what you want. You quarrel and fight. You do not have, because you do not ask God. 3 When you ask, you do not receive, because you ask with wrong motives, that you may spend what you get on your pleasures.

There are some wonderful prayer promises in the Gospel of John. Note the Promises and the Conditions:
John 14:13 I will do whatever you ask in my name, so that the Son may bring glory to the Father.
John 14:14 You may ask me for anything in my name, and I will do it.
John 15:7 If you remain in me and my words remain in you, ask whatever you wish, and it will be given you.
John 15:16 ... I … appointed you to go and bear fruit … the Father will give you whatever you ask in my name.
John 16:23 ... I tell you the truth, my Father will give you whatever you ask in my name.
John 16:24 Until now you have not asked for anything in my name. Ask and you will receive, and your joy will be complete.
John 16:26-27 In that day you will ask in my name. I am not saying that I will ask the Father on your behalf. No, the Father himself loves you because you have loved me and have believed that I came from God. (See 1 Jn 3:22; 5:14-16)

MK 11:22 "Have faith in God," Jesus answered. 23 "I tell you the truth, if anyone says to this mountain, `Go, throw yourself into the sea,' and does not doubt in his heart but believes that what he says will happen, it will be done for him. 24 Therefore I tell you, whatever you ask for in prayer, believe that you have received it, and it will be yours.