Atonement is an important idea in Scripture. "Atonement" is really an Old Testament word, but the principle is only fully developed in the New Testament. This isn't unusual in Scripture: a great deal of the teaching in the New Testament is rooted in the Old Testament. We remember the Lord Jesus' words to Nicodemus: "Thou art the teacher of Israel and knowest not these things" (John 3:10, DBY). Not everything in the New Testament is in the Old, but the New Testament is certainly based in the Old.
Atonement is most fully developed in Leviticus 16, the instructions for the great Day of Atonement. This was the one day a year the high priest was allowed to come into the Holy of Holies.
Speak unto Aaron thy brother, that he come not at all times into the sanctuary inside the veil before the mercy-seat which is upon the ark, that he die not... And this shall be an everlasting statute unto you, to make atonement for the children of Israel to cleanse them from all their sins once a year.
Leviticus 16:2, 34 (DBY)
The Day of Atonement consisted of several offerings: it started with a young bullock and a ram (Leviticus 16:3). The bullock was a sin offering, the ram was to be a burnt offering. The bullock was to be offered as a sin-offering, to make atonement for the high priest "and for his house" (Leviticus 16:6 & 11, DBY). Hebrews 9:6–14 makes a point of contrasting the Day of Atonement with Christ's priesthood in Heaven in the present day. He hasn't entered with the blood of goats and calves (Hebrews 9:12), but by His own blood.
Let's think about that a minute: in Leviticus 16, the high priest needs an offering for himself, because he is a sinner. But Scripture doesn't say that he goes in only for himself: he has to take in the offering for himself "and for his house". Christ entered into Heaven with His own blood, but we know He didn't need a sacrifice for His own sins. He had no sin, He knew no sin, and He did no sin. But He brought in an offering "for His house".
But let's consider the rest of what Leviticus 16 prescribes. After the priest offered the bullock as an sin offering for himself and his house, there were two goats. Lots were to be cast over the goats, one goat was to be a sin-offering, the other was "for Azazel" (Leviticus 16:7–10, DBY).
The goat for the sin-offering was to be killed, and its blood was to be taken into the Holy of Holies (Leviticus 16:15–19). The blood was to be sprinkled on the mercy-seat, before the mercy-seat, and on the altar. He was to "make atonement for the sanctuary" and "for the tent of meeting" (Leviticus 16:16). The priest was to "make atonement for himself, and for his house, and for the whole congregation of Israel" (Leviticus 16:17). He was to make atonement for the altar too (Leviticus 16:18).
The King James Version renders the "goat for Azazel" as the "scapegoat", which is probably easier to understand. This goat wasn't to be killed as an offering. This goat was brought to Aaron, and Aaron was to put his hands on the goat's head and confess all the sins of the people over the goat. Then the goat was to be led into the desert, "that the goat may bear upon him all their iniquities to a land apart" (Leviticus 16:22).
Now it's interesting that both goats are said to make atonement. The scapegoat makes atonement, "the goat upon which the lot fell for Azazel shall be set alive before Jehovah, to make atonement with it" (Leviticus 16:10, DBY). The goat for the sin-offering makes atonement with its blood (Leviticus 16:15–17). The two goats make atonement in two very different ways, but they both make atonement. So atonement has two parts: propitiation and substitution.
Propitiation is made when the blood is presented to God. To "propitate" is "to make favorably inclined" (Dictionary.com). The New Testament talks about propitiation, it says "[Christ] is the propitiation for our sins; but not for ours alone, but also for the whole world" (1 John 2:2, DBY). When Aaron went in to the Holy of Holies, he was to make atonement for himself, his house, the people, and even the Tabernacle itself. Propitiation is a God-ward truth. It's not really about us, it's about how God views us. Propitiation is God looking favorably on us.
Propitiation is linked to blood. This is an important principle: "it is the blood that maketh atonement for the soul" (Leviticus 17:11, DBY). This is what the goat for the sin-offering teaches. It is blood presented to God that makes atonement for the people. Christ has made atonement for us in His own blood.
Substitution is when our sins are taken away, put on someone else. The scapegoat carried away the sins of the people into "a land apart". This is what Peter talks about: Christ "who himself bore our sins in his body on the tree" (1 Peter 2:24, DBY). Christ was our Substitute when He took our sins as though they were His own.
We notice a difference between the two parts of atonement: there is confession involved on the part of the people when it comes to substitution, but not when it comes to propitiation. In a sense, propitiation is really between the priest and the Lord. There's no confession of sins: it's not individual sins that are in view at all.
The New Testament connects these two ideas to Christ's blood and His body. It was in His body that He bore our sins (1 Peter 2:24), but it was in His blood that He made peace (Colossians 1:20). So we might say that substitution speaks to us of Christ's body, propitiation speaks to us of His blood.
There is a long debate between Christians about the nature and scope of atonement. Is the atonement Christ made for us universal? does it apply to all people? Or is it limited, does it only apply to the redeemed? Well, it's both, isn't it? Christ is the Propitiation for the whole world; in Leviticus, atonement was made even for inanimate objects. But substitution is a lot more particular: Christ didn't bear the sins of the whole world in His body on the tree, He bore our sins. Propitiation doesn't require confession, substitution does.
J. N. Darby wrote an excellent article on atonement called, "Propitiation and Substitution". He argues that "arminians" understand propitiation (not substitution), while "calvinists" see substitution (not propitiation). It's well worth a read.
Atonement has two parts, Propitiation and Substitution. There's a lot more to be learned from Leviticus 16: we're really only hitting the high points. But Christ gave Himself for us: both His body and His blood. He bore our sins in His own body; He made peace in His blood. An appreciation of the atonement really needs to involve an appreciation of both truths.
propitiate. (n.d.). Online Etymology Dictionary. Retrieved August 20, 2013, from Dictionary.com website: http://dictionary.reference.com/browse/propitiate
Darby, J. N., "Propitiation and Substitution", Collected Writings of J. N. Darby (Vol. 27, p. 318), Believers Bookshelf, Sunbury PA, 1971, from STEM Publishing website: http://stempublishing.com/authors/darby/EXPOSIT/27034E.html