Many people have attempted to understand the doctrine of the trinity.  They have created analogies and metaphors that work on some levels, but never fully explain it.  This is likely because the concept of one God being revealed in three separate persons is above our current understanding.  However, simply because we cannot fully understand the trinity now and simply because the word trinity isn't in the scriptures, it doesn't mean the triune Godhead does not exist.

We are told in the scriptures that God's thoughts are higher than our thoughts and His ways higher than our ways.  We cannot fully understand God, Jesus Christ, or the Holy Spirit.  It follows suit that we cannot completely understand the trinity.

This article shows the scriptures in the Bible about the trinity.  There are many verses that reveal God the Father, God the Son, and God the Holy Spirit as one God, revealed in three separate persons (the same in character, different in function).

1. There is One God

Old Testament: Deut. 4:35; 39; 32:39; 2 Sam. 22:32; Isa. 37:20: 43:10; 44:6-8; 45:5; 14; 21-22; 46:9

New Testament: John 5:44; Rom. 3:30; 16:27; 1 Cor. 8:4-6; Gal. 3:20; Eph. 4:6; 1 Tim. 1:17; 2:5; James 2:19; Jude 25

2. Plural Pronouns For God

Gen. 1:26; 3:22; 11:7; Isa. 6:8

3. Jesus Christ's Father is God

John 17:3; 1 Cor. 8:6; 2 Cor. 1:3; Eph. 1:3; 1 Pet. 1:3

4. Jesus Christ is God

Isa. 9:6; John 1:1; John 20:28; Acts 20:28; Rom. 9:5; Titus 2:13; Heb. 1:8; 2 Pet. 1:1; 1 John 5:20

5. Jesus is Jehovah

Rom. 10:9-13; Phil. 2:9-11; Heb. 1:10; 1 Pet. 2:34; 1 Pet. 3:14-15; Deut. 6:4; Mark 12:29; 1 Cor. 8:6; Eph. 4:5; cf. Roma. 10:12; 1 Cor. 12:5

6. Jesus Uses God's Titles

1. The first and the last: Rev. 1:17; 22:13; cf. Isa. 44:6
2. King of kings and Lord of lords: 1 Tim. 6:15; Rev. 17:14; 19:16
3. Savior: Luke 2:11; John 4:42; 1 John 4:14; Titus 2:13, cf. v. 10; etc.; cf. Isa. 43.11; 45:21-22; 1 Tim 4:10
5. Shepherd: John 10:11; Heb. 13:20; cf. Psa. 23:1; Isa. 40:11
6. Rock: 1 Cor. 10:4; cf. Isa. 44:8

7. Jesus Received Honors Reserved for God Alone

1. Honor: John 5:23
2. Love: Matt. 10:37
3. Prayer: John 14:14; Acts 7:59-60 (cf. Luke 23:34, 46); Rom. 10:12-13; 1 Cor. 1:2; 2 Cor. 12:8-10 (where "the Lord" must be Jesus, cf. v. 9), etc.
4. Worship (proskuneo): Matt. 28:17; Heb. 1:6 (cf. Psa. 97:7); cf. Matt 4:10
5. Religious or sacred service (latreuo): Rev. 22:13
6. Faith: John 3:16; 14:1

8. Jesus Does the Works of God

1. Creation: John 1:3; 1 Cor. 8:6; Col. 1:16-17; Heb. 1:2; Rev. 3:14 (cf. 21:6 on "beginning"); on "through" and "in" cf. Rom. 11:36; Heb. 2:10; Acts 17:28; cf. also Isa. 44:24
2. Sustains the universe: Col. 1:17; Heb. 1:3
3. Forgives sins: Matt. 9:1-8; Mark 2:1-12; Luke 5:17-26; note that Jesus forgives sins not committed against Him; all of them: John 5:17-29 (including judgment, cf. Matt. 25:31-46; 2 Cor. 5:10)

9. Jesus Has All the Incommunicable Attributes of God

1. All of them: John 1:1; Col. 1:15; 2:9; Heb. 1:3
2. Self-existent: John 5:26
3. Unchangeable: Heb. 1:10-12; 13:8
4. Eternal: John 1:1; 8:58; 17:5; Col. 1:17; Heb. 1:2
5. Omnipresent: Matt. 18:20; 28:20; John 3:13; Eph. 1:23; 4:10' Col. 3:11
6. Omniscient: John 16:30; 21:17; cf. 2:23-24
7. Incomprehensible: Matt. 11:25-27

10. Jesus is the Son of God

1. The phrase "Son of God" used in the gospels: Matt. 4:3; 4:6; 8:29; 14:33; 26:63; 27:40; 27:43; 27:54; Mark 1:1; 3:11; 15:39; Luke 1:35; 3:38; 4:3; 4:9; 4:41; 22:70; John 1:34; 1:49; 3:18; 5:25; 9:35; 10:36; 11:4; 11:27; 19:7; 20:31
2. The phrase "Son of God" used in the rest of the New Testament:  Acts 8:37; 9:20; Rom. 1:4; 2 Cor. 1:19; Gal. 2:20; Eph. 4:13; Heb. 4:14; 6:6; 7:3; 10:29; 1 John 3:8; 4:15; 5:5; 5:10; 5:12; 5:13; 5:20; Rev. 2:18
3. Equal with God: John 5:17-23; 8:58-59; 10:30-39; 19:7; Matt. 26:63-65 (Jesus' repeated claim to be the Son of God was always understood by the Jewish leaders as a blasphemous claim to equality with God, an interpretation Jesus never denied)
4. Jesus is therefore God's Son, not God's creation, God's servant, God's agent, etc.; Jesus is God's Son who became a servant for our sake and for the Father's glory (John 13:13-15; 17:4; Phil. 2:6-11; Heb. 1:4-13; 3:1-6; 5:8; etc.)

11. The Holy Spirit is God

1. Equated with God: Acts 5:3-4; 2 Cor. 3:17-18

2. Has the incommunicable attributes of God
A. Eternal: Heb. 9:14
B. Omnipresent: Psa. 139:7
C. Omniscient: 1 Cor. 2:10-11

3. Involved in all the works of God
A. Creation: Gen. 1:2; Psa. 104:30
B. Incarnation: Matt. 1:18, 20; Luke 1:35
C. Resurrection: Rom. 1:4; 8:11
D. Salvation: Rom. 8:1-27

4. Is a person
A. Has a name: Matt. 28:19; note that even though "name" might be used of a nonperson, here, in conjunction with the Father and the Son, it must be used of a person
B. Is the "Helper"
C. Is another Helper: John 14:16, cf. 1 John 2:1; note also that "Helper" (parakletos) was used in Greek always or almost always of persons
D. Is sent in Jesus' name, to teach: John 14:26
E. Will arrive, and then bear witness: John 15:26-27
F. Is sent by Christ to convict of sin, will speak not on his own but on behalf of Christ, will glorify Christ, thus exhibiting humility: John 16:7-14
G. Is the Holy Spirit, in contrast to unholy spirits: Mark 3:22-230, cf. Matt. 12:32; 1 Tim. 4:1; 1 John 3:24-4:6
H. Speaks, is quoted as speaking: John 16:13; Acts 1:16; 8:29; 10:19; 11:12; 13:2; 16:6; 20:23; 21:11: 28:25-27; 1 Tim. 4:1; Heb. 3:7-11; 10:15-17; 1 Pet. 1:11; Rev. 2:7, 11, 17, 29; 3:6, 13,22
I. Can be lied to: Acts 5:3
J. Can make decisions, judgments: Acts 15:28
K. Intercedes for Christians with the Father: Rom. 8:26
L. "Impersonal" language used of the Spirit paralleled by language used of other persons
M. The Holy Spirit as fire: Matt. 3:11; Luke 3:16; cf. Ex. 3:2-4; Deut. 4:24; 9:3; Heb. 12:29
N. The Holy Spirit poured out: Acts 2:17, 33; cf. Isa. 53:12; Phil. 2:17; 2 Tim. 4:6
O. Being filled with the Holy Spirit: Eph. 5:18, etc.; cf. Eph. 3:17, 19; 14:10

12. The Father, the Son, and the Holy Spirit are Distinct Persons

A. Matt. 28:19: "the Father and the Son and the Holy Spirit": use of definite article before each personal noun indicates distinct persons unless explicitly stated otherwise; compare Rev. 1:17; 2:8, 26

The views that "Father" and "Son" are distinct persons but not the Holy Spirit, or that the Holy Spirit is not a person at all, or that all three are different offices or roles of one person, are impossible in view of the grammar (together with the fact that in Scripture a "spirit" is a person unless context shows otherwise).

Does singular "name" prove that the three are one person? No; cf. Gen. 5:2; 11:14; 48:6; and esp. 48:16
"Name" need not be personal name, may be title: Isa. 9:6; Matt. 1:23. If a single personal name is sought, the name shared by all three persons is "Yahweh" or "Jehovah."

B. Acts 2:38 and Matt. 28:19
Neither passage specifies that certain words are to be spoken during baptism; nor does the Bible ever record someone saying, "I baptize you in the name of...."

Those said to be baptized in the name of Jesus (whether or not the formula "in the name of Jesus" was used) were people already familiar with the God of the OT:
Jews: Acts 2:5, 38; 22:16
Samaritans: Acts 8:5, 12, 16
God-fearing Gentiles: Acts 10:1-2, 22, 48
Disciples of John the Baptist: Acts 19:1-5
The first Christians in Corinth were Jews and God-fearing Gentiles: Acts 18:1-8; 1 Cor. 1:13
Trinitarian formula for baptism (if that is what Matt. 28:19 is) was given in context of commissioning apostles to take the gospel to "all the nations," including people who did not know of the biblical God

God the Father and the Son Jesus Christ are two persons

A. The salutations: Rom. 1:7; 1 Cor. 1:3; 2 Cor. 1:2; Gal. 1:3; Eph. 1:2; 6:23; Phil. 1:2; 1 Thess. 1:1; 2 Thess. 1:1, 2; 1 Tim. 1:1, 2; 2 Tim. 1:2; Tit. 1:4; Phm. 3; James 1:1; 2 Pet. 1:2; 2 John 3
B. Two witnesses: John 5:31-32; 8:16-18; cf. Num. 35:30; Deut. 17:6; 19:15
C. The Father sent the Son: John 3:16-17; Gal. 4:4; 1 John 4:10; etc.; cf. John 1:6; 17:18; 20:21
D. The Father and the Son love each other: John 3:35; 5:20; 14:31; 15:9; 17-23-26; cf. Matt. 3:17 par.; 17:5 par.; 2 Pet. 1:17
E. The Father speaks to the Son, and the Son speaks to the Father: John 11:41-42; 12:28; 17:1-26; etc.
F. The Father knows the Son, and the Son knows the Father: Matt. 11:27; Luke 10:22; John 7:29; 8:55; 10:15
G. Jesus our Advocate with the Father: 1 John 2:1

Jesus is not God the Father

A. Isa. 9:6: "Father of eternity" means eternal; compare other names formed with word "father": Abialbon, "father of strength" = strong (2 Sam. 23:31); Abiasaph, "father of gathering" = gatherer (Ex. 6:24); Abigail, a woman's name(!), "father of exultation" = exulting (1 Chron. 2:16)
B. John 10:30: Jesus did not say, "I am the Father," nor did He say, "the Son and the Father are one person."
The first person plural esmen ("we are") implies two persons
The neuter word for "one" (hen) is used, implying essential unity but not personal unity (compare John 17:21-23).
C. John 5:43: Jesus' coming in His Father's name means not that He was the Father because He had the Father's name, but that, while others come in their own name (or their own authority), Jesus does not; He comes in His Father's name (on His Father's authority)
John 8:19; 16:3: Ignorance of Jesus is indeed ignorance of the Father, but that does not prove that Jesus is the one He calls "My Father"
D. John 14:6-11: Jesus and the Father are one being, not one person
Jesus said, "I am in the Father," not "I am the Father"
The statement, "the Father is in Me," does not mean Jesus is the Father; compare John 14:20; 17:21-23
E. John 14:18: An older adult brother can care for his younger siblings, thus preventing them from being "orphans," without being their father
F. Colossians 2:9: Does not mean that Jesus is the Father, or that Jesus is an incarnation of the Father; rather, since "Godhead" (theotes) means Deity, the state of being God, the nature of God, Jesus is fully God, but not the only person who is God. "The Godhead" here does not = the Father (note that Jesus is in the Father, John 10:38; 14:10, 11; 17:21), but the nature of the Father.
The Father and the Son are both involved in various activities: raising Jesus (Gal. 1:1; John 2:19-22), raising the dead (John 5:21); 6:39-40, 44, 54, 1 Cor. 6:14), answering prayer (John 14:13-14; 15:16; 16:23), sending the Holy Spirit (John 14:16; 15:26; 16:7), drawing people to Jesus (John 6:44; 12:32), etc. These common works do prove that the two persons are both God, but not that Jesus is the Father

The Son existed before His Incarnation, even before creation

Prov. 30:4: This is not predictive prophecy; "prophecy" in 30:1 translates massa, which is rendered elsewhere as "burden."
The Son created all things: See VI.E.1
Jesus was "with" (pros or para) God the Father before creation: John 1:1; 17:5; pros in John 1:1 does not mean "pertaining to," although is does in Hebrews 2:17; 5:1 (which use pros with ta).
Jesus, the Son of God, existed before John the Baptist (who was born before Jesus): john 1:15, cf. 1:14-18, 29-34
Jesus, the Son, came down from heaven, sent from the Father, and went back to heaven, back to the Father: John 3:13, 31; 6:33; 38, 41, 46, 51, 56-58, 62; 8:23, 42; 13:3; 16:27-28; cf. Acts 1:10-11; cf. the sending of the Holy Spirit, John 16:5-7; 1 Pet. 1:12
Jesus, speaking as the Son (John 8:54-56), asserts His eternal preexistence before Abraham: John 8:58
The Son explicitly said to exist "before all things": Col. 1:17, cf. 1:12-20
These statements cannot be dismissed as true only in God's foreknowledge
We are all in God's min before creation; yet such passages as John 1:1 and John 17:5 clearly mean to say something unusual about Christ.
To say that all things were created through Christ means that He must have existed at creation.
No one else in Scripture is ever said to have been with God before creation.
Texts which speak of the Son being begotten "today" do not mean He became the Son on a certain day, since they refer to His exaltation at the resurrection (Acts 13:33; Heb. 1:3-5; 5:5; cf. Psa. 2:7; cf. also Rom. 1:4)

Jesus is not the Holy Spirit

The Holy Spirit is "another Comforter": John 14:16; compare 1 John 2:1
Jesus sent the Holy Spirit: John 15:26; 16:7
The Holy Spirit exhibits humility in relation to, and seeks to glorify, Jesus (John 16:13-14)
The Son and the Holy Spirit are distinguished as tow persons in Matt. 28:19
The Holy Spirit descended upon Jesus: Luke 3:22
Is Jesus the Holy Spirit?
2 Cor. 3:17: the Spirit is here called "Lord" in the sense of being Yahweh or God, not Jesus (cf. v. 16, citing Ex. 34:34; cf. v. 17 in the New English Bible); note Acts 28:25-27, cf. Isa. 6:8-10
1 Cor. 15:45: Jesus is "a life-giving Spirit," not in the sense that He is the Holy Spirit whom He sent at Pentecost, but in the sense that He is the glorified God-man; and as God He is Spirit by nature. All three persons of the Trinity are Spirit, though there are not three divine Spirits; and only one person is designated "the Holy Spirit."
Rom. 8:27, 34: the fact that two persons intercede for us is consistent with the fact that we have two Advocates (John 14:16; Rom. 8:26; 1 John 2:1)
John 14:18: Jesus here refers to His appearances to the disciples after the resurrection (compare 14:19), not to the coming of the Spirit
Jesus and the Holy Spirit are both involved in various activities: raising Jesus (John 2:19-19-22); Rom. 8:9-11), raising the dead (John 5:21; 6:39-40, 44, 54, Rom. 8:9-11), dwelling in the believer (John 14:16; 2 Cor. 13:5; Col. 1:27), interceding for the believer (Rom. 8:26; Heb. 7:25), sanctifying believers (Eph. 5:26; 1 Pet. 1:2), etc. These works prove that the two persons are both God, but not that Jesus is the Holy Spirit.

The Father is not the Holy Spirit

The Father sent the Holy Spirit: John 14:15; 15:26
The Holy Spirit intercedes with the Father for us: Rom. 8:26-27
The Father and the Holy Spirit are distinguished as two persons in Matt. 28:19
Is the Father the Holy Spirit?
Matt. 1:18; Luke 1:35: It is argued that the Holy Spirit is the Father of the incarnate Son of God; this argument ignores the fact that the "conception" is not a product of physical union between a man and a woman!
The Father and the Holy Spirit are both said to be active in various activities; the resurrection of Jesus (Gal. 1:1; Rom. 8:11), comforting Christians (2 Cor. 1:3-4; John 14:26), sanctifying Christians (Jude 1; 1 Pet. 1:2), etc. The most these facts prove is that the two work together; they do not prove the two are one person.

13.  Conclusion: The Bible teaches the Trinity

All the elements of the doctrine are taught in Scripture:
One God
The Father is God
The Son is God
The Holy Spirit is God
The Father, Son, and Holy Spirit are three persons (i.e., they are not each other, nor are they impersonal; they relate to one another personally)
The New Testament presents a consistent triad of Father, Son, Holy Spirit (God, Christ, Spirit): matt. 28:19; 2 Cor. 13:34; also Luke 1:35; 3:21-22 par.; 4:1-12; John 4:10-25; 7:37-39; 14-16; 20:21-22; Acts 1:4-8; 2:33, 38-39; 5:3-4, 9, 30-32; 7:55-56; 10:36-38, 44-48; 11:15-18; 15:8-11; 20:38; 28:25-31; Rom. 1:1-4; 5:5-10; 8:2-4, 9-11, 14-17; 1 Cor. 6:11; 12:4-6, 11-12, 18; 2 Cor. 1:19-22; 3:6-8, 14-18; Gal. 3:8-14; 4:4-7; Eph. 1:3-17; 2:18, 21-22; 3:14-19; 4:4-6, 29-32; 5:18-20; Phil. 3:3; 1 Thess. 1:3-6; 2 Thess. 2:13-14; Tit. 3:4-6; Heb. 2:3-4; 9:14; 10:28-31; 1 Pet. 1:2; 1 John 3:21-24; 4:13-14; Jude 20-21; Rev. 2:18, 27-29
Therefore, the Bible does teach the Trinity

13. What Difference Does The Doctrine Of The Trinity Make?

Sovereignty: Because the three persons have each other, we can be assured that God created us only to share the love they have and not as a means to His own end: Acts 17:25; John 17:21-26
Mystery: The triune God is totally unlike anything in our world, and therefore greater than anything we can comprehend: Rom. 11:33-36; Isa. 40:18
Salvation: God alone planned our salvation, came to save us, and dwells in us to complete our salvation: 1 Pet. 1:2; Eph. 1:3-18; etc.
Prayer: We pray to the Father through the Son, and also pray to the Son directly, in the Spirit: John 14:13-14; Eph. 2:18; etc.
Worship: We worship Father and Son in the Spirit: john 4:23-24; Phil. 3:3; Heb. 1:8; etc.
Love: The love among the three persons is the basis and model for our love for one another: John 17:26
Unity: The unity of the three persons is the basis and model for the unity of the church: John 17:21-23
Humility: As the persons of the Trinity seek the glory of each other, so we should seek the interests of others above our own: Phil. 2:5-11; John 16:13-14
Sonship: We are "sons of God" as we are united with the Son of God by the work of the Holy Spirit and the adoption of the Father: john 1:12-23; Rom. 8:14-17
Truth: All those who wish to worship and love God must seek to know Him as He is in truth, for God, as Father, Son, and Holy Spirit, is truth: John 4:24; 14:6, 17; 15:26; 16:13

Citation: Much of this was from Robert Bowman's article on the trinity.