God reveals Himself as Father, Son and Holy Spirit. So the Holy Spirit is a part of God.
The Holy Spirit was involved in the creation of the world.
Now the earth was formless and empty,
darkness was over the surface of the deep,
and the Spirit of God was hovering over the waters.
Genesis 1:2 (NIV)
The Holy Spirit was sent by God the Father to live inside Jesus when he walked on the earth.
Jesus, full of the Holy Spirit,
returned from the Jordan and was led by the Spirit in the desert,
Luke 4:1 (NIV)
Jesus promised that the Holy Spirit would be given to other people too.
On the last and greatest day of the Feast, Jesus stood and said in a loud voice, “If anyone is thirsty, let him come to me and drink. Whoever believes in me, as the Scripture has said, streams of living water will flow from within him.” By this he meant the Spirit, whom those who believed in him were later to receive. Up to that time the Spirit had not been given, since Jesus had not yet been glorified. On hearing his words, some of the people said, “Surely this man is the Prophet.”
John 7:37-40 (NIV)
God has promised to give His Holy Spirit to all Christians.
Peter replied, “Repent and be baptised, every one of you, in the name of Jesus Christ for the forgiveness of your sins. And you will receive the gift of the Holy Spirit. The promise is for you and your children and for all who are far off—for all whom the Lord our God will call.”
Acts 2:38-39 (NIV)