The ProblemThe problem is sin. Genesis 2:16-17 "And the LORD God said to the man, saying, 'You may surely eat of every tree of the garden, but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die.'" Then, when Adam and Eve do eat of that fruit, God gives the first pointer to Christ when he curses the serpent, 3:15 "I will put enmity between you and the woman, and between your offspring and her offspring; he shall bruise your head, and you shall bruise his heel." Even clearer, Genesis 6:5-7 "The LORD saw that the wickedness of man was great in the earth, and that every intention of the thoughts of his heart was only evil continually. And the LORD was sorry that he had made man on the earth, and it grieved him to his heart. So the LORD said, 'I will blot out man whom I have created from the face of the land, man and animals and creeping things and birds of the heavens, for I am sorry that I have made them.'" Here we see clearly God's hatred of sin such that he desires that it be destroyed, this is his righteousness. This can be seen again in Genesis 18 and 19, where God destroys Sodom and Gomorrah because "their sin is very grave" Genesis 18:20. Isaiah 1:27-28 "Zion shall be redeemed by justice, and those in her who repent, by righteousness. But rebels and sinners shall be broken together, and those who forsake the LORD shall be consumed." Yet we cannot of ourselves be righteous, "As it is written: 'none is righteous, no, not one; no one understands; no one seeks for God. All have turned aside; together they have become worthless; no one does good, not even one.'" Romans 3:10-12, and, verse 20, "For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin" Further, we cannot bear this ourselves, "For the wages of sin is death," Romans 6:23.
So, the problem is that we carry the weight of sin, and God is righteous and cannot, therefore, bear to allow sin to remain in his sight. The purpose of the atonement, then, is to remove that sin apart from blotting out all of humanity, which he would be justified in doing. Thus John the baptist calls Jesus "the Lamb of God, who takes away the sin of the world!" when he sees him, John 1:29.
Requirements of the AtonementFrom the problem, we can see certain requirements which God has implied are required of any solution to the problem of sin. First: there must be a sacrifice, as the old testament has sacrifices for sins. "But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God" Hebrews 10:12.
Second, the sins must be removed from us, and placed on this sacrifice, who is Christ. If the sin remains on us, then we still must bear it. This is the same as it was when animal sacrifices were offered, for "And Aaron shall lay both his hands on the head of the live goat, and confess over it all the iniquities of the people of Israel, and all their transgressions, all their sins. And he shall put them on the head of the goat and send it away into the wilderness by the hand of a man who is in readiness. The goat shall bear all their iniquities on itself to a remote area, and he shall let the goat go free in the wilderness." Leviticus 16:21-22. But, rather, "Christ redeemed us from the curse of the law by becoming a curse for us--for it is written, 'Cursed is everyone who is hanged on a tree.'" and, 1 Peter 2:24 "He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed." And, similarly, Romans 6:5- "For if we have been united with him in a death like his, we shall certainly be united with him in a resurrection like his. We know that our old self was crucified with him in order that the body of sin might be brought to nothing, so that we would no longer be enslaved to sin." This is how God may forgive us, because in Christ's death, the suffering due sin has been paid, so that the wrath of God against our sin has been satisfied. It is not as though there is some well from which to draw for forgiveness of sins, but rather the sins themselves were laid upon Christ such that he bore the punishment due to each of them. It is by Christ's death that we are reconciled, Romans 5:10, so that there is nothing we need to do to gain that reconciliation, but if we had to go and ask for that forgiveness, then we would be utterly lost, since there would be no reconciliation by which we might draw near to ask for that forgiveness we so desperately need. It is sin that keeps us apart from God, so that if we had to ask that he remove our sin, he would not hear us, for he does not hear the prayers of the wicked, which we certainly are if we are still in sin, which is precisely why we need this reconciliation.
Third, the atonement is the defeat of Satan, that is, it destroys Satan's power over those it is effectual for, since it was promised that "he shall bruise your head." Thus Romans 6:17 "But thanks be to God, that you who were once slaves of sin have become obedient from the heart to the standard of teaching to which you were committed." From which flows, Colossians 2:6, "Therefore, as you received Christ Jesus the Lord, so walk in him," that is, since Christ not only bore the penalty for our sin but bruised the head of the enemy, and since we, in him, die to sin, and as sin is thus dead to us, walk in that life which you have as one who is risen with Christ and therefore not under the power of sin, just as you are no longer under the guilt of sin.
Fourth, the atonement was costly. That is, we had a debt which Christ payed, or again, Christ ransomed us. This does not seem to me to flow quite so neatly from the above problem, however, Numbers 18:15 "Everything that opens the womb of all flesh, whether man or beast, which they offer to the LORD, shall be yours. Nevertheless, the firstborn of man you shall redeem, and the firstborn of unclean animals you shall redeem." Here is a redeeming, i.e., a lamb is put in the place of a man. Psalm 111:9 "He sent redemption to his people; he has commanded his covenant forever. Holy and awesome is his name!" And yet, Psalm 49:7-9 "Truly no man can ransom another, or give to God the price of his life, for the ransom of their life is costly and can never suffice, that he should live on forever and never see the pit." Thus the necessity of a man who is without blemish to be the ransom, and verse 15 "But God will ransom my soul from the power of Sheol, for he will receive me." 1 Peter 1:18-19 "knowing that you were ransomed from the futile ways inherited from your forefathers, not with perishable things such as silver or gold, but with the precious blood of Christ, like that of a lamb without blemish or spot." Thus we have the picture of Christ being the lamb who comes as the ransom who takes our place as the perfect sacrifice.