There are different theories of atonement that attempt to explain how humans can be reconciled to God, or how God reconciles us to him.
Atonement can mean to wipe out or to cover. Their is also a connection between the terms "kofer" (ransom) and "Kapper" to cover. The following theories for the most part agree that it is God who does something to make a way for people to be brought into a relationship with Him. The different theories hold different ways of explaining how God did it.
Moral InfluenceThe moral influence view of the atonement teaches that the purpose and work of Jesus Christ was to bring positive moral change to humanity. This moral change came through the teachings and example of Jesus, the Christian movement he founded, and the inspiring effect of his martyrdom and resurrection.
The Biblical support for this theory comes from passages where Jesus and His followers give instruction on how to live better, how to love better, how to become the people God intended them to be.
According to this view, people were held in bondage or captivity to sin or Satan. Christ's sacrifice is seen as a ransom being paid to Satan (or the hold sin has on us) to set people free. Redeeming in this case means buying back. An analogy would be warriors being held captive by another kingdom, and the home king giving something to the other kingdom to get his warriors back.
Biblical support would be Matthew 20:28, Mark 10:45, 1 Timothy 2:6, Galatians 3:13, Galatians 4:5, 1 Peter 1:18-19, Revelation 5:9, Romans 3:23-26, and Titus 2:14 which speak in terms of redeeming and ransom. 1 Corinthians 7:23, Colossians 1:13 speak of rescuing. 1 Corinthians 6:20, 1 Corinthians 7:23 - speak of being bought with a price.
Christus VictorThis theory speaks to Christ defeating the powers of darkness. The view holds that humanity was under the authority of sin or Satan since the fall of Adam. Jesus strikes a defeating blow to Satan when he defeats sin and death.
Biblical support: John 12:31, John 16:11, 2 Timothy 1:10, Hebrews 2:14-15, 1 John 3:8
Biblical support: Isaiah 53, 1 Peter 3:18, Romans 3:23-26
Satisfaction and Penal substitution atonement
There are a variety of theories within these two theories, and to me they seem to overlap.
There is the idea of God's justice being satisfied. God's character demands that all sin must be accounted for. Since the the wages of sin is death, and mankind is sinful, someone must die. God's justice is only satisfied if either we are punished for our sins, or Jesus dies as our substitute. Jesus' death is seen as a payment to God for the debt of sin.
There are parts of each of these atonement theories that are meaningful to me.
I'm hesitant to agree with the 'moral influence' theory, since it may lean more towards human works than God's working. But I do see some truth in the fact that Jesus did have a lot to say on how to live and follow His example.
To me it seems like the ransom theory makes a lot of sense to me. The idea of Jesus buying us back from slavery makes sense to me, and looks to me like it has a lot of Scriptural support.
I also see truth in the Christus Victor theory. The idea that the powers of darkness were defeated at the cross sounds right as well.
I also agree that the wages of sin is death, and than mankind falls short of God's holiness. However I have some concerns with aspects of this theory as I have written about before here.
But yes, as Christians have tried to sort out the mysteries of how God made things right with us there have been different theories used to explain it. People have studied our Scriptures and found different ways to understand this deep mystery. I tend to lean in favour of some of the theories more than others. I respect your right to study the different theories and determine what makes the most sense to you. We may differ on the 'how' God makes things right, but I trust we can agree that God does make things right.