Excerpt from Concise Theology by J.I. Packer.
The Old Testament constantly insists that there is only one God, the self-revealed Creator, who must be worshiped and loved exclusively (Deut. 6:4-5; Isa. 44:6– 45:25). The New Testament agrees (Mark 12:29-30; 1 Cor. 8:4; Eph. 4:6; 1 Tim. 2:5) but speaks of three personal agents, Father, Son, and Holy Spirit, working together in the manner of a team to bring about salvation (Rom. 8; Eph. 1:3-14; 2 Thess. 2:13-14; 1 Pet. 1:2). The historic formulation of the Trinity (derived from the Latin word trinitas, meaning “threeness”) seeks to circumscribe and safeguard this mystery (not explain it; that is beyond us), and it confronts us with perhaps the most difficult thought that the human mind has ever been asked to handle. It is not easy; but it is true.
The doctrine springs from the facts that the New Testament historians report, and from the revelatory teaching that, humanly speaking, grew out of these facts. Jesus, who prayed to his Father and taught his disciples to do the same, convinced them that he was personally divine, and belief in his divinity and in the rightness of offering him worship and prayer is basic to New Testament faith (John 20:28-31; cf. 1:18; Acts 7:59; Rom. 9:5; 10:9-13; 2 Cor. 12:7-9; Phil. 2:5-6; Col. 1:15-17; 2:9; Heb. 1:1-12; 1 Pet. 3:15). Jesus promised to send another Paraclete (he himself having been the first one), and Paraclete signifies a many-sided personal ministry as counselor, advocate, helper, comforter, ally, supporter (John 14:16-17, 26; 15:26-27; 16:7-15). This other Paraclete, who came at Pentecost to fulfill this promised ministry, was the Holy Spirit, recognized from the start as a third divine person: to lie to him, said Peter not long after Pentecost, is to lie to God (Acts 5:3-4).
So Christ prescribed baptism “in the name (singular: one God, one name) of the Father and of the Son and of the Holy Spirit”—the three persons who are the one God to whom Christians commit themselves (Matt. 28:19). So we meet the three persons in the account of Jesus’ own baptism: the Father acknowledged the Son, and the Spirit showed his presence in the Son’s life and ministry (Mark 1:9-11). So we read the trinitarian blessing of 2 Corinthians 13:14, and the prayer for grace and peace from the Father, the Spirit, and Jesus Christ in Revelation 1:4-5 (would John have put the Spirit between the Father and the Son if he had not regarded the Spirit as divine in the same sense as they are?). These are some of the more striking examples of the trinitarian outlook and emphasis of the New Testament. Though the technical language of historic trinitarianism is not found there, trinitarian faith and thinking are present throughout its pages, and in that sense the Trinity must be acknowledged as a biblical doctrine: an eternal truth about God which, though never explicit in the Old Testament, is plain and clear in the New.
The basic assertion of this doctrine is that the unity of the one God is complex. The three personal “subsistences” (as they are called) are coequal and coeternal centers of self-awareness, each being “I” in relation to two who are “you” and each partaking of the full divine essence (the “stuff” of deity, if we may dare to call it that) along with the other two. They are not three roles played by one person (that is modalism), nor are they three gods in a cluster (that is tritheism); the one God (“he”) is also, and equally, “they,” and “they” are always together and always cooperating, with the Father initiating, the Son complying, and the Spirit executing the will of both, which is his will also. This is the truth about God that was revealed through the words and works of Jesus, and that undergirds the reality of salvation as the New Testament sets it forth.
The practical importance of the doctrine of the Trinity is that it requires us to pay equal attention, and give equal honor, to all three persons in the unity of their gracious ministry to us. That ministry is the subject matter of the gospel, which, as Jesus’ conversation with Nicodemus shows, cannot be stated without bringing in their distinct roles in God’s plan of grace (John 3:1-15; note especially vv. 3, 5-8, 13-15, and John’s expository comments, which NIV renders as part of the conversation itself, vv. 16-21). All non-Trinitarian formulations of the Christian message are by biblical standards inadequate and indeed fundamentally false, and will naturally tend to pull Christian lives out of shape.