One God, three persons
The word Trinity does not appear in the Bible, but Christians use it to summarise what Scripture shows about God. There is one God, and the Father, the Son, and the Holy Spirit each share in who God is. Jesus gave his disciples one baptismal name that includes all three: the Father, the Son, and the Holy Ghost (Matthew 28:19).
A person, in this sense, is not a human body or a separate god. It means someone who knows, wills, speaks, and relates. The Father is not the Son, the Son is not the Spirit, and the Spirit is not the Father. Yet Christians worship one God, not three gods. It is a mystery, but not a contradiction.
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
“Third” does not mean lesser
When people say “the third person”, they are usually following the familiar order, Father, Son, Holy Spirit. The number does not rank the Spirit below the other two. He was not created after the Father and Son, and he is not God's junior helper.
The Bible treats the Holy Spirit as God. When Ananias lied about his gift in Jerusalem, Peter said he had lied to the Holy Ghost and also said he had lied to God (Acts 5:3-4). Paul also describes the Spirit as knowing the deep things of God (1 Corinthians 2:10). These are not descriptions of a mere influence or energy.
But Peter said, Ananias, why hath Satan filled thine heart to lie to the Holy Ghost, and to keep back part of the price of the land? Whiles it remained, was it not thine own? and after it was sold, was it not in thine own power? why hast thou conceived this thing in thine heart? thou hast not lied unto men, but unto God.
The Spirit is personal, not a force
Sometimes Scripture speaks of people being filled with the Spirit or receiving power through the Spirit. That can make it sound as if the Holy Spirit is like electricity. But the Spirit does things a personal force cannot do. He teaches believers, reminds them of Jesus' words, and is sent by the Father in Jesus' name (John 14:26).
He can also be grieved. Paul tells Christians in Ephesus not to grieve the Holy Spirit of God (Ephesians 4:30). Paul says the Spirit helps believers in weakness and intercedes when they do not know how to pray as they ought (Romans 8:26). This is personal care, not an impersonal spiritual feeling.
The Spirit brings us to Jesus
The Spirit's work is deeply connected to the Father and the Son. Jesus promised that the Father would give his followers another Comforter, or Helper, to remain with them (John 14:16-17). The Spirit makes Christ known, strengthens God's people, convicts hearts of truth, and helps them live as God's children.
So this teaching is not only a puzzle about theology. It means God is near. Through the Holy Spirit, Christians are not left to follow Jesus by determination alone. Read John 14 slowly, and notice how Jesus speaks of the Father's love, his own presence, and the Spirit's help together.
And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever; Even the Spirit of truth; whom the world cannot receive, because it seeth him not, neither knoweth him: but ye know him; for he dwelleth with you, and shall be in you.