The expression "to make atonement" is frequent in Exodus, Leviticus, and Numbers, but rare in the rest of the Bible. The basic idea, however, is widespread. The need for it arises from the fact that man is a sinner, a truth made plain throughout Scripture but infrequent outside the Bible.
In the OT sin is dealt with by the offering of sacrifice. Thus the burnt offering will be accepted "to make atonement" (Lev. 1:4), as also the sin offering and the guilt offering (Lev. 4:20; 7:7) and especially the sacrifices on the day of atonement (Lev. 16). Of course, sacrifice is ineffective if offered in the wrong spirit. To sin "with a high hand" (Num. 15:30), i.e., proudly and presumptuously, is to place oneself outside the sphere of God's forgiveness. The prophets have many denunciations of the offering of sacrifice as the expression of a repentant and trustful heart is to find atonement. Atonement is sometimes made by means other than the sacrifices, such as the payment of money (Exod. 30:12-16) or the offering of life (II Sam. 21:3-6). In such cases to make atonement means "to avert punishment, especially the divine anger, by the payment of a koper, a ransom,' which may be of money or which may be of life" (L. Morris, The Apostolic Preaching of the Cross, 166). Throughout the OT sin is serious; it will be punished unless atonement is sought in the way God has provided.
This truth is repeated and enlarged upon in the NT. There it is made clear that all men are sinners (Rom. 3:23) and that hell awaits them (Mark 9:43; Luke 12:5). But it is just as clear that God wills to bring salvation and that he has brought it in the life, death, resurrection, and ascension of his Son. The love of God is the mainspring (John 3:16; Rom. 5:8). We are not to think of a loving Son as wringing salvation from a just but stern Father. It is the will of the Father that men be saved, and salvation is accomplished, not with a wave of the hand, so to speak, but by what God has done in Christ: "God was in Christ reconciling the world to himself" (II Cor. 5:19), a reconciliation brought about by the death of Christ (Rom. 5:10). The NT emphasizes his death, and it is no accident that the cross has come to be accepted as the symbol of the Christian faith or that words like "crux" and "crucial" have come to have the significance that they possess. The cross is absolutely central to salvation as the NT sees it. This is distinctive of Christianity. Other religions have their martyrs, but the death of Jesus was not that of a martyr. It was that of a Savior. His death saves men from their sins. Christ took their place and died their death (Mark 10:45; II Cor. 5:21), the culmination of a ministry in which he consistently made himself one with sinners.
The NT does not put forward a theory of atonement, but there are several indications of the principle on which atonement is effected. Thus sacrifice must be offered, not the sacrifice of animals, which cannot avail for men (Heb. 10:4), but the perfect sacrifice of Christ (Heb. 9:26; 10:5-10). Christ paid sin's due penalty (Rom. 3:25-26; 6:23; Gal. 3:13). He redeemed us (Eph. 1:7), paying the price that sets us free (I Cor. 6:20; Gal. 5:1). He made a new covenant (Heb. 9:15). He won the victory (I Cor. 15:55-57). He effected the propitiation that turns away the warth of God (Rom. 3:25), made the reconciliation that turns enemies into friends (Eph. 2:16). His love and his patient endurance of suffering set an example (I Pet. 2:21); we are to take up our cross (Luke 9:23). Salvation is many-sided. But however it is viewed, Christ has taken our place, doing for us what we could not do for ourselves. Our part is simply to respond in repentance, faith, and selfless living.
(Elwell Evangelical Dictionary)