THE word "Prayer" expresses the largest and most comprehensive way to approach God. It gives importance to the ingredient of devotion. It is a relationship and interaction with God. It is enjoyment of God. It is access to God. "Supplication" is a more controlled and more intense form of prayer, accompanied by a sense of personal need, limited to the seeking in an urgent manner the supply for a pressing need.
"Supplication" is the very soul of prayer in the way of pleading for some one thing, very much needed, and the need intensely felt.
"Intercession" is amplification in prayer; it is going out in broadness and fullness from ones self for others. Primarily, it does not center in praying for others, but refers to the freeness, boldness and childlike confidence in praying. It is the completeness of trusting influence in the soul's approach to God, unlimited and unhesitating in its access and its demands. This influence and confident trust is to be used in prayer for others.
Prayer is always and everywhere an immediate and confiding approach to, and a request of, God the Father. In the prayer universal and perfect "The Lord's Prayer" [or as I like to refer to this prayer as "The Family Prayer"], as the pattern of all praying, it is "Our Father, Who art in Heaven." At the grave of Lazarus, Jesus lifted up His eyes and said, "Father." In His pastoral prayer, Jesus lifted up His eyes to Heaven, and said, "Father." Personal, familiar and fatherly was always His way of praying. Strong, instrumental, touching and tearful, was also His way of praying. Read these words of Paul: "Who in the days of His flesh, when He had offered up prayers and supplications, with strong crying and tears, unto Him that was able to save Him from death, and was heard in that He feared" (Hebrews 5:7).
So it is the same elsewhere in the Holy Bible (James 1:5) we have "asking" put into view as a prayer: "If any of you lack wisdom, let him ask of God, who gives to all men liberally, and upbraiding [scolding somebody] not, and it shall be given him."
"Asking of God" and "receiving" from the Lord - direct request to God, immediate connection with God - that is true prayer.
The Holy Bible says in (John 5:13) this statement about prayer: "And this is the confidence that we have in Him, that if we ask anything according to His will, He hears us. And if we know that He hears us, whatsoever we ask, we know that we have the petitions that we desired of Him."
In (Phil. 4:6) we have these words about prayer: "Be careful for nothing, but in everything, by prayer and supplication, with thanksgiving, let your requests be made known unto God."