What is the Holy Ghost?

Response to comment [from an atheist]:  "I realize that this is about the trinity and therefore will be disputed as long as Christianity exists, though this Holy Ghost thing has me baffled. I get the belief in God and understand why there are differences with Jesus is God and Jesus is the son of God disputes. Though how does the Holy Ghost fit into the picture. What manifestation does the Holy Ghost take within Christianity?"

He's the one you've grieved (Eph 4:30). Hope he hasn't given up on you (Lk 19:44).

This
song was written to him (Jn 14:17).

What is the Holy Ghost?