what is the nature of the holy spirit?
Sometimes referred to as the 'Holy Ghost',
this third person of the Trinity is perhaps the least
understood. These mysterious names might help contribute to
the confusion, but they are intended to actually help our
understanding. Most importantly, the Holy Spirit is God and
therefore has all the nature of God.
The Old Testament word for Spirit (ruach) is the word for breath, wind, or spirit. In that language and understanding, the spirit was the breath that keeps us alive (because we breathe when we are alive, and we don't breathe when we're dead.) The Spirit is the source of life, in this way of thinking. And according to the creation account, God's Spirit breathed life into the nostrils of man, causing him to be alive (Genesis 2:7). This is the likeness of God that we possess - a spirit, that was given to us from God. It is not a 'piece' of God, but is our own spirit, which is dependent upon God for life, and allows us to have conscious relationship with the Spirit of God.
So to summarize, God speaks to us, to our spirit, through His Spirit. It is through the Holy Spirit that God lives in us, and brings Jesus the Christ to us.
Confused? That's ok. Again, we would expect the nature of the transcendent God to transcend our knowledge. We just have to accept what He has described for us to understand. The important point to know about the Holy Spirit is that He IS GOD, manifest in a spirit that speaks to us through our spirit. He is one of the three persons of the Trinity (see more about the Trinity and the nature of God for further clarifications), and He has specific duties (see more about His duties in "What Does The Holy Spirit Do?").