Note: Click the Greek word for more in depth study.
Ask (154) (aiteo) means to ask for with urgency, even to the point of demanding. Aiteo more frequently suggests attitude of a suppliant (one who supplicates [supplicate is from Latin supplex = bowed] means to makes a humble, earnest plea or entreaty), the petitioning of one who is lesser in position than he to whom the petition is made. To ask means to to call on for an answer, which indicates that we believe there is someone (our Father) listening. It also implies that we expect Him to answer or otherwise why ask?
Beg (1189) (deomai from deo = to bind) means to ask for something with the sense of pleading, beseeching or begging. To request, to ask for with a sense of urgency and a presumed need. When used in the context of prayer deomai means to make petition, to plead, to ask in prayer, to implore (pray for earnestly) and emphasizes the existence of a need. Deomai is a strong way to ask for something - a leper imploring Jesus to heal him (Lk 5:12), a father's desperate plea to Jesus to cast a demon out of his son (Lk 9:38).
Pray (4336) (proseuchomai from pros = toward, facing, before [emphasizing the direct approach of the one who prays in seeking God’s face] + euchomai = originally to speak out, utter aloud, express a wish, then to pray or to vow. Greek technical term for invoking a deity) in the NT is always used of prayer addressed to God (to Him as the object of faith and the One who will answer one’s prayer) and means to speak consciously (with or without vocalization) to Him, with a definite aim. Proseuchomai encompasses all the aspects of prayer -- submission, confession, petition, supplication (may concern one's own need), intercession (concerned with the needs of others), praise, and thanksgiving. Vine says that proseuchomai carries with it a notion of worship (but see the Greek word for worship = proskuneo) which is not present in the other words for prayer (eg, aiteo, deomai, both of which involve spoken supplication)
Pray (2172)(euchomai; euche = a vow in Acts 18:18, a prayer in Jas 5:15) literally meant to speak out or utter aloud and came to mean as used by to express a wish. The idea can be to desire something, with the implication of a pious wish. To vow (this sense is primarily found in the Septuagint uses but is found in Acts 26:29). To pray, but in the NT the derivative proseuchomai is the more common verb for pray.Euchomai and proseuchomai are more general terms for prayer, while deomai conveys the sense of seeking to have a need met.
Prayer (1162) (deesis) refers to urgent requests or supplications to meet a need and are exclusively addressed to God. Deesis prayers arise from one's sense of need (which reflects a humble heart) and in knowing what is lacking. This individual's plea is in turn made to God to supply for the need. Deesis in the New Testament always carries the idea of genuine entreaty and supplication before God. It implies a realization of need and a petition for its supply. In Classical Greek deesis (in contrast to the Biblical uses) was not restricted to sacred uses, but was employed of requests preferred to men.
Prayer (4335)(proseuche from pros = toward or immediately before + euchomai = to pray or vow) is the more general word for prayer and is used only of prayer to God. The prefix pros would convey the sense of being immediately before Him and hence the ideas of adoration, devotion, and worship. The basic idea is to bring something, and in prayer this pertains to bringing up prayer requests. In early Greek culture an offering was brought with a prayer that it be accepted. Later the idea was changed slightly, so that the thing brought to God was a prayer. In later Greek, prayers appealed to God for His presence. Proseuche is used 37 times (click) in the NT. Note the concentration of prayer in the early church! (Acts) What has happened to us as a church in America? How might this relate to how infrequently we see the power of the Lord at work in our midst? On the positive side what might happen if all the believers in America began to intercede for Spirit wrought, Word centered, Christ exalting, God glorifying revival???