The Holy Spirit is God's continual presence inside of us.
In the Old Testament, He empowers and sustains Israel's leaders. In the New Testament, He empowers and sustains the whole Church.
In the OT:
- the Spirit was involved in creation: an extension of God's power that was present on Earth (Gn. 1.2)
- He worked in leaders and prophets, to establish, deliver, judge, guide, and restore the people of God. Leaders were endowed with the Spirit for the tasks they did.
- He was the Spirit of prophecy
- He inspired the writings of the OT Scriptures
In the NT:
- He lived in Jesus and empowered Him for ministry.
- After the resurrection, the Spirit is freed to be the Spirit of Christianity as a permanent arrangement.
- He is the functional equivalent of Jesus on Earth
- He is a Counselor and guide for Christians
- He advocates our causes before the Father
- He prosecutes the world, showing people their sin and convicting them of it.
- He makes sure there is always a testimony about Jesus on Earth.
- He guides people into truth about God and Jesus.
- He inspired the writings of the NT.
Jesus states His oneness with the Spirit in John 14.16-18. He is put on an equal par with the Father and Son in Matthew 28.19; 2 Cor. 13.14 et al. Hopefully that helps. We can talk about this more if you want.