Trinity means Tri-Unity - A group consist of three closely related members. The union of three divine persons, the Father, Son and Holy Spirit as one God. In other words God in three persons. This is similar to a human made of Body, Soul and Spirit.
For simpler understanding of biblical truth about Trinity in a diagram:
Why Trinity is important? Following taken from Billy Graham Association Website Q&A section:
The Bible shows very clearly that there is only one God, and yet that there are three personal distinctions in His complex nature, traditionally referred to as "three Persons in the Godhead"—God the Father, God the Son, and God the Holy Spirit.
Each is distinct from the others but never acts independently. They are one in nature and purpose. This mystery is called the doctrine of the Trinity, though that term is not used in the Bible. The teaching, however, is present in seed form in the Old Testament and is revealed explicitly in the New Testament. Note passages such as Matthew 28:19; John 10:30, 14:26; 2 Corinthians 13:14.
Our finite minds cannot understand or explain this mystery of God, which is nevertheless a fact. We must accept the truths found in the Word of God by faith even though we ourselves cannot comprehend them fully; read Hebrews 11:1, 3, 6 and 1 Corinthians 2:5-10, 14; 13:12.
It is really not surprising that the infinite God should be complex in His nature beyond the ability of finite humans to comprehend! This doctrine is absolutely essential to New Testament Christianity. Theologians have pointed out that if it were not true, the Bible would be unreliable, Christ would not be divine, and His death on the cross would not atone for our sins, being merely the death of a martyr.
Following are supporting verses for Trinity:
Genesis 1:26 - Then God (means Elohim in Hebrew - 2 or more - Father, Jesus and Holy Spirit. It is plural form of El means God) said, "Let Us make man in Our image, according to Our likeness; let them have dominion over the fish of the sea, over the birds of the air, and over the cattle, over all the earth and over every creeping thing that creeps on the earth."
Deut 6:4 - 4 "Hear, O Israel: The Lord our God, the Lord is one! 5 You shall love the Lord your God with all your heart, with all your soul, and with all your strength. 6 "And these words which I command you today shall be in your heart. 7 You shall teach them diligently to your children, and shall talk of them when you sit in your house, when you walk by the way, when you lie down, and when you rise up.
Matthew 3:16 - 13 Then Jesus came from Galilee to John at the Jordan to be baptized by him. 14 And John tried to prevent Him, saying, "I need to be baptized by You, and are You coming to me?" 15 But Jesus answered and said to him, "Permit it to be so now, for thus it is fitting for us to fulfill all righteousness." Then he allowed Him. 16 When He had been baptized, Jesus came up immediately from the water; and behold, the heavens were opened to Him, and He saw the Spirit of God descending like a dove and alighting upon Him. 17 And suddenly a voice came from heaven, saying, "This is My beloved Son, in whom I am well pleased."
Matthew 28:19 - 19 Go therefore and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 teaching them to observe all things that I have commanded you; and lo, I am with you always, even to the end of the age." Amen.
Luke 1:35 - And the angel answered and said to her, "The Holy Spirit will come upon you, and the power of the Highest will overshadow you; therefore, also, that Holy One who is to be born will be called the Son of God.
Luke 10:21 - In that hour Jesus rejoiced in the Spirit and said, "I thank You, Father, Lord of heaven and earth, that You have hidden these things from the wise and prudent and revealed them to babes. Even so, Father, for so it seemed good in Your sight.
Luke 23:46 - And when Jesus had cried out with a loud voice, He said, "Father, 'into Your hands I commit My spirit.' " Having said this, He breathed His last.
John 1:33 - I did not know Him, but He who sent me to baptize with water said to me, 'Upon whom you see the Spirit descending, and remaining on Him, this is He who baptizes with the Holy Spirit.'
John 8:16 - 13 The Pharisees therefore said to Him, "You bear witness of Yourself; Your witness is not true." 14 Jesus answered and said to them, "Even if I bear witness of Myself, My witness is true, for I know where I came from and where I am going; but you do not know where I come from and where I am going. 15 You judge according to the flesh; I judge no one. 16 And yet if I do judge, My judgment is true; for I am not alone, but I am with the Father who sent Me. 17 It is also written in your law that the testimony of two men is true. 18 I am One who bears witness of Myself, and the Father who sent Me bears witness of Me."
John 10:30 - I and My Father are one.
John 14:16 - And I will pray the Father, and He will give you another Helper, that He may abide with you forever
John 14:26 - But the Helper, the Holy Spirit, whom the Father will send in My name, He will teach you all things, and bring to your remembrance all things that I said to you.
John 16:13-15 - 13 However, when He, the Spirit of truth, has come, He will guide you into all truth; for He will not speak on His own authority, but whatever He hears He will speak; and He will tell you things to come. 14 He will glorify Me, for He will take of what is Mine and declare it to you. 15 All things that the Father has are Mine. Therefore I said that He will take of Mine and declare it to you.
John 17:5 - And now, O Father, glorify Me together with Yourself, with the glory which I had with You before the world was.
John 17:24 - Father, I desire that they also whom You gave Me may be with Me where I am, that they may behold My glory which You have given Me; for You loved Me before the foundation of the world.
John 20:21-22 - 21 So Jesus said to them again, "Peace to you! As the Father has sent Me, I also send you." 22 And when He had said this, He breathed on them, and said to them, "Receive the Holy Spirit.
Acts 2:33-38 - 33 Therefore being exalted to the right hand of God, and having received from the Father the promise of the Holy Spirit, He poured out this which you now see and hear. 34 For David did not ascend into the heavens, but he says himself: 'The Lord said to my Lord, "Sit at My right hand, 35 Till I make Your enemies Your footstool." ' 36 "Therefore let all the house of Israel know assuredly that God has made this Jesus, whom you crucified, both Lord and Christ." 37 Now when they heard this, they were cut to the heart, and said to Peter and the rest of the apostles, "Men and brethren, what shall we do?" 38 Then Peter said to them, "Repent, and let every one of you be baptized in the name of Jesus Christ for the remission of sins; and you shall receive the gift of the Holy Spirit
Acts 7:55 - But he, being full of the Holy Spirit, gazed into heaven and saw the glory of God, and Jesus standing at the right hand of God,
Acts 10:38 - how God anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him.
Romans 15:16 - that I might be a minister of Jesus Christ to the Gentiles, ministering the gospel of God, that the offering of the Gentiles might be acceptable, sanctified by the Holy Spirit.
1 Corinthians 2:5-10, 14 - 5 that your faith should not be in the wisdom of men but in the power of God. 6 However, we speak wisdom among those who are mature, yet not the wisdom of this age, nor of the rulers of this age, who are coming to nothing. 7 But we speak the wisdom of God in a mystery, the hidden wisdom which God ordained before the ages for our glory, 8 which none of the rulers of this age knew; for had they known, they would not have crucified the Lord of glory. 9 But as it is written: "Eye has not seen, nor ear heard, Nor have entered into the heart of man The things which God has prepared for those who love Him." 10 But God has revealed them to us through His Spirit. For the Spirit searches all things, yes, the deep things of God. 11 For what man knows the things of a man except the spirit of the man which is in him? Even so no one knows the things of God except the Spirit of God. 12 Now we have received, not the spirit of the world, but the Spirit who is from God, that we might know the things that have been freely given to us by God. 13 These things we also speak, not in words which man's wisdom teaches but which the Holy Spirit teaches, comparing spiritual things with spiritual. 14 But the natural man does not receive the things of the Spirit of God, for they are foolishness to him; nor can he know them, because they are spiritually discerned.
1 Corinthians 13:12 - 12 For now we see in a mirror, dimly, but then face to face. Now I know in part, but then I shall know just as I also am known. 13 And now abide faith, hope, love, these three; but the greatest of these is love.
2 Corinthians 1:21-23 - 21 Now He who establishes us with you in Christ and has anointed us is God, 22 who also has sealed us and given us the Spirit in our hearts as a guarantee. 23 Moreover I call God as witness against my soul, that to spare you I came no more to Corinth.
2 Corinthians 13:14 - The grace of the Lord Jesus Christ, and the love of God, and the communion of the Holy Spirit be with you all. Amen.
Gal 4:6 - And because you are sons, God has sent forth the Spirit of His Son into your hearts, crying out, "Abba, Father!"
Eph 3:14-16 - 14 For this reason I bow my knees to the Father of our Lord Jesus Christ, 15 from whom the whole family in heaven and earth is named, 16 that He would grant you, according to the riches of His glory, to be strengthened with might through His Spirit in the inner man,
Eph 4:4-6 - 4 There is one body and one Spirit, just as you were called in one hope of your calling; 5 one Lord, one faith, one baptism; 6 one God and Father of all, who is above all, and through all, and in you all
1 Thessalonians 5:23 - Now may the God of peace Himself sanctify you completely; and may your whole spirit, soul, and body be preserved blameless at the coming of our Lord Jesus Christ.
Hebrews 11:1, 3, 6 - 1 Now faith is the substance of things hoped for, the evidence of things not seen. 2 For by it the elders obtained a good testimony. 3 By faith we understand that the worlds were framed by the word of God, so that the things which are seen were not made of things which are visible. 4 By faith Abel offered to God a more excellent sacrifice than Cain, through which he obtained witness that he was righteous, God testifying of his gifts; and through it he being dead still speaks. 5 By faith Enoch was taken away so that he did not see death, "and was not found, because God had taken him"; for before he was taken he had this testimony, that he pleased God. 6 But without faith it is impossible to please Him, for he who comes to God must believe that He is, and that He is a rewarder of those who diligently seek Him.
1 Peter 1:2 - elect according to the foreknowledge of God the Father, in sanctification of the Spirit, for obedience and sprinkling of the blood of Jesus Christ: Grace to you and peace be multiplied.
1 John 5:7 - 4 For whatever is born of God overcomes the world. And this is the victory that has overcome the world--our faith. 5 Who is he who overcomes the world, but he who believes that Jesus is the Son of God? 6 This is He who came by water and blood--Jesus Christ; not only by water, but by water and blood. And it is the Spirit who bears witness, because the Spirit is truth. 7 For there are three that bear witness in heaven: the Father, the Word, and the Holy Spirit; and these three are one. 8 And there are three that bear witness on earth: the Spirit, the water, and the blood; and these three agree as one.