Recapitulation - Irenaeus (ca. 130-202) Christ recapitulated in himself all the stages of human life that related to sin. In this way he reversed the course on which Adam, by his sin, started humanity.
Ransom to Satan - Origen (ca. 185-254) Christ's death was a ransom paid to Satan for claims he had on man.
Satisfaction - Anselm (1033-1109) Christ's death rendered satisfaction to God's honor.
Substitutionary (Reformers, from Anselm ) that Jesus Christ suffered crucifixion as a substitute for human sin, satisfying God's just wrath against man's transgression due to Christ's infinite merit.
Moral Influence - Abelard (1079-1142) Christ's death was a manifestation of God's love. The suffering love of Christ awakens a responsive love in sinners.
Example - Socinus (1539-1604) Christ's death did not atone for sin. By his teaching in life and
example in death, Christ brought salvation to man.
Governmental - Hugo Grotius (1583-1645) Sin disrupted God's government. By his death Christ demonstrated the high estimate God placed on his law and government.
Mystical - Friedrich Schleiermacher (1768-1834) Christ's death exercises some influence to change man. Christ's unbroken unity with God enabled him to bring a potential mystical influence for good to man through his death.
Vicarious Repentance - McLeod Campbell (1800-1872) By his death Christ offered to God a perfect and vicarious repentance which man could not perform but from which he benefits.
There are two main points to salvation, both accomplished by God Himself, and we the benefactors.
Christ lived a righteous life on our behalf. This was a NECESSITY. Christ could have been born, God could have had Him stuck down by Herod, and hey an atonement, but that's not what happened, Isaiah 53 pretty much explains the gospel, which is remarkable, no wonder it's called the 5th Gospel. When John said "It is you who should be baptizing me", Jesus said "No it must be so that I FULFILL ALL RIGHTEOUSNESS", what? Christ had to fulfill all righteousness, for Him? No, FOR US.
Two, Christ died for the elects sin. "I lay down my life for my sheep". "I pray NOT FOR THE WORLD, but for those you gave me". On the cross, the great fulfillment of all the glorious OT prophecies, (Sorry getting excited), God placed the sins of the elect on Christ, which seems unjust except when we understand Jesus volunteered to take the punishment for our sins. "But wait one objects, our punishment is eternal, Jesus suffered some three hours under the Fathers wrath". True, but He IS ETERNAL, 3 hours was sufficient for Him to feel the grief of each and everyone of our sins.
If Jesus merely removed our sins, we'd be back to square one, like Adam, innocent without sin, but Paul EXHAUSTS THE SUBJECT in Romans, we, in addition to have no sins to our account, are given a "Foreign Righteousness", God applies through the Holy Spirit into each believer Christs righteousness.
What now? Now, for me, all my sins are paid for, AND Christs righteous life is imputed, accounted to me. It's called the "Great Exchange", our sins to Christ, His righteousness to us, if you are a believer. If you believe not your sin remains, and you will be judged on that great day.
Look at my moniker and read it, so true and GLORIOUS! Man, on the side of the devil, living in rebellion to God, yet while we were sinning, God, who had no need of man, man whose only deserts are destruction, sent His son out of glory into the form of human flesh, learn and live a sinless life, suffer more than any man in his life, being that pure and seeing rebellion everywhere, being tested in all ways yet without sin, then going through with it, oh PRAISE PRAISE PRAISE GOD, He took upon His body our sins, God judged them, AND THEY WILL NEVER BE JUDGED AGAIN, then God does what the OT Jews couldn't even look upon, the glory of God, put into our hearts, how can we be condemned? By Christ? He died for us, by the Father? He put our sins on His son, by the Spirit? No He lives in us and is our advocate, we are truly free if we are in Christ, FREE. And though now under the Angelic realm, our brother and King Jesus is bringing us up to His realm, it's just so glorious, so amazing, why are we not rejoicing in this alone, always?
IT IS COMPLETE, DONE, now if you are in Christ go live for Him, thank Him, try to glorify Him, forget your sins, focus on Christ and if you sin confess it to HIM, not to some sick priest, and go on. Just have faith in God, love your brothers and you will live, and NEVER COME INTO CONDEMNATION