We're not explicitly told who or what the Holy Spirit is in scripture. However, we are given clues, which can help us to better understand and explain Him.
In Ephesians 4:30, we're told that He is a spiritual being who can be grieved. This reveals a lot in itself; the Holy Spirit is a person and has emotion. According to Romans 15:30, we see He is loving, possessing the same love of the Father since they have the same nature, and in Psalms 139:7-10 we also see He is omnipresent. We know the Holy Spirit has the same nature of God and is a member of the triune God as implicitly stated in Matthew 28:19.
We're also told the Holy Spirit is an advocate, teacher, brings freedom, gives both wisdom and understanding, convicts / guides, restrains evil and empowers believers. (John 14:26, 2 Corinthians 3:17, Isaiah 11:2, John 16:8, Romans 8:14, 2 Thessalonians 2:7, Acts of the Apostles 1:8). This is just my personal belief, but I do believe the power of the Father and Son comes from the Holy Spirit Himself, such as when the universe was created.
Every member of the Trinity seems to have their own unique personality and role, but all have the same nature. I see them as being similar to a family unit. The Father is the highest authority, and glorifies His Son. Jesus, the Son, is the agent of the Father, and seeks to glorify His Father. The Holy Spirit is the main acting force behind everything, an "invisible mover", and also glorifies the Son.
I also like VMae's description, regarding God having mind, body and spirit. Perhaps the Father would be the mind (the one who commands, just like how our mind issues commands to our body), the Son Jesus would be the body (the Word made flesh) and the Holy Spirit would be God's own spirit. This actually makes a lot of sense.