The most difficult thing about the Christian concept of the Trinity is that there is no way to adequately explain it. The Trinity is a concept that is impossible for any human being to fully understand, let alone explain. God is infinitely greater than we are, therefore we should not expect to be able to fully understand Him. The Bible teaches that the Father is God, that Jesus is God, and that the Holy Spirit is God. The Bible also teaches that there is only one God. Though we can understand some facts about the relationship of the different persons of the Trinity to one another, ultimately, it is incomprehensible to the human mind. However, this does not mean it is not true or not based on the teachings of the Bible.
Keep in mind when studying this subject that the word "Trinity" is not used in Scripture. This is a term that is used to attempt to describe the triune God, the fact that there are 3 coexistent, co-eternal persons that make up God. Understand that this is NOT in any way suggesting 3 Gods. The Trinity is 1 God made up of 3 persons. There is nothing wrong with using the term "Trinity" even though the word is not found in the Bible. It is shorter to say the word "Trinity" than to say "3 coexistent, co-eternal persons making up 1 God." If this presents a problem to you, consider this: the word grandfather is not used in the Bible either. Yet, we know there were grandfathers in the Bible. Abraham was the grandfather of Jacob. So don't get hung up on the term "Trinity" itself. What should be of real importance is that the concept that is REPRESENTED by the word "Trinity" does exist in Scripture. With the introduction out of the way, Bible verses will be given in discussion of the Trinity.
1) There is one God: Deuteronomy 6:4; 1 Corinthians 8:4; Galatians 3:20; 1 Timothy 2:5.
2) The Trinity consists of three Persons: Genesis 1:1; 1:26; 3:22; 11:7; Isaiah 6:8; 48:16; 61:1; Matthew 3:16-17; Matt 28:19; 2 Corinthians 13:14. In the passages in the Old Testament, a knowledge of Hebrew is helpful. In Genesis 1:1, the plural noun "Elohim" is used. In Genesis 1:26; 3:22; 11:7 and Isaiah 6:8, the plural pronoun for "us" is used. That "Elohim" and "us" refer to more than two is WITHOUT question. In English, you only have two forms, singular and plural. In Hebrew, you have three forms: singular, dual, and plural. Dual is for two ONLY. In Hebrew, the dual form is used for things that come in pairs like eyes, ears, and hands. The word "Elohim" and the pronoun "us" are plural forms - definitely more than two - and must be referring to three or more (Father, Son, Holy Spirit).
In Isaiah 48:16 and 61:1, the Son is speaking while making reference to the Father and the Holy Spirit. Compare Isaiah 61:1 to Luke 4:14-19 to see that it is the Son speaking. Matthew 3:16-17 describes the event of Jesus' baptism. Seen in this is God the Holy Spirit descending on God the Son while God the Father proclaims His pleasure in the Son. Matthew 28:19 and 2 Corinthians 13:14 are examples of 3 distinct persons in the Trinity.
3) The members of the Trinity are distinguished one from another in various passages: In the Old Testament, "LORD" is distinguished from "Lord" (Genesis 19:24; Hosea 1:4). The "LORD" has a "Son" (Psalm 2:7, 12; Proverbs 30:2-4). Spirit is distinguished from the "LORD" (Numbers 27:18) and from "God" (Psalm 51:10-12). God the Son is distinguished from God the Father (Psalm 45:6-7; Hebrews 1:8-9). In the New Testament, John 14:16-17 is where Jesus speaks to the Father about sending a Helper, the Holy Spirit. This shows that Jesus did not consider Himself to be the Father or the Holy Spirit. Consider also all of the other times in the Gospels where Jesus speaks to the Father. Was He speaking to Himself? No. He spoke to another person in the Trinity - the Father.
4) Each member of the Trinity is God: The Father is God: John 6:27; Romans 1:7; 1 Peter 1:2. The Son is God: John 1:1, 14; Romans 9:5; Colossians 2:9; Hebrews 1:8; 1 John 5:20. The Holy Spirit is God: Acts 5:3-4; 1 Corinthians 3:16 (The One who indwells is the Holy Spirit - Romans 8:9; John 14:16-17; Acts 2:1-4).
5) The subordination within the Trinity: Scripture shows that the Holy Spirit is subordinate to the Father and the Son, and the Son is subordinate to the Father. This is an internal relationship, and does not deny the deity of any person of the Trinity. This is simply an area which our finite minds cannot understand concerning the infinite God. Concerning the Son see: Luke 22:42; John 5:36; John 20:21; 1 John 4:14. Concerning the Holy Spirit see: John 14:16; 14:26; 15:26; 16:7 and especially John 16:13-14.
6) The tasks of the individual members of the Trinity: The Father is the ultimate source or cause of: 1) the universe (1 Corinthians 8:6; Revelation 4:11); 2) divine revelation (Revelation 1:1); 3) salvation (John 3:16-17); and 4) Jesus' human works (John 5:17; 14:10). The Father INITIATES all of these things.
The Son is the agent through whom the Father does the following works: 1) the creation and maintenance of the universe (1 Corinthians 8:6; John 1:3; Colossians 1:16-17); 2) divine revelation (John 1:1; Matthew 11:27; John 16:12-15; Revelation 1:1); and 3) salvation (2 Corinthians 5:19; Matthew 1:21; John 4:42). The Father does all these things through the Son, who functions as His agent.
The Holy Spirit is the means by whom the Father does the following works: 1) creation and maintenance of the universe (Genesis 1:2; Job 26:13; Psalm 104:30); 2) divine revelation (John 16:12-15; Ephesians 3:5; 2 Peter 1:21); 3) salvation (John 3:6; Titus 3:5; 1 Peter 1:2); and 4) Jesus' works (Isaiah 61:1; Acts 10:38). Thus the Father does all these things by the power of the Holy Spirit.
None of the popular illustrations are completely accurate descriptions of the Trinity. The egg (or apple) fails in that the shell, white, and yolk are parts of the egg, not the egg in themselves. The Father, Son and Holy Spirit are not parts of God, each of them is God. The water illustration is somewhat better but still fails to adequately describe the Trinity. Liquid, vapor, and ice are forms of water. The Father, Son, and Holy Spirit are not forms of God, each of them is God. So, while these illustrations may give us a picture of the Trinity, the picture is not entirely accurate. An infinite God cannot be fully described by a finite illustration. Instead of focusing on the Trinity, try to focus on the fact of God's greatness and infinitely higher nature than our own. "Oh, the depth of the riches of the wisdom and knowledge of God! How unsearchable his judgments, and his paths beyond tracing out! Who has known the mind of the Lord? Or who has been his counselor?" (Romans 11:33-34)