I think it can vary depending on denomination but within the belief of the trinity, it is one god in the form of three persons (father, son, holy spirit). I am certainly no theologian, though, so take that with a grain of salt.
I want to preface that I am agnostic at this point in my life and I struggle with a good understanding of a great example of these and mostly the inconsistencies of their descriptions throughout denominations. But I think in general the "holy spirit" is supposed to be the "feelings" you get that "draw you into" God, or for some people the "callings" they seem to have.
Christians believe that Jesus' is atonement makes it possible for them to be resurrected and to repent and be forgiven so they can return to their Heavenly Father's presence.