Most Christians know that Jesus died on the Cross for our sins. Often misunderstood is the concept of Atonement. The term is tied to the idea that Jesus was a substitute for us in the forgiveness of our sins. This idea is throughout the New Testament. According to N. Batzig, Jesus only explicitly tied those aspects of His death on the cross to its meaning on three occasions; For even the Son of Man did not come to be served, but to serve, and to give his life for many. (Mark 10:45), in the Good Shepherd discourse (John 10) and at the institution of the Lord’s Supper (Luke 22:19–20). In these places, Jesus taught the substitutionary nature of His death for the forgiveness of the sins of His people. When we move from the Gospels to the Epistles, an explicit articulation of the substitutionary nature of the death of Christ appears. When one considers the many instances in which the Apostles explain the death of Christ, it is incontrovertible that the doctrine of substitutionary atonement is the Apostolic doctrine of the atonement. In what is perhaps the clearest exposition of the death of Christ, the Apostle Paul teaches the vicarious sacrifice of the Savior when he declares, “For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God” (2 Cor. 5:21). Likewise, the Apostle Peter explained that Jesus “himself bore our sins in his body on the tree, that we might die to sin and live to righteousness” (1 Peter 2:24).
Behind the Apostolic interpretation of the death of the Savior is the Old Testament teaching on the atonement. In Leviticus 16:21-22, we read;"Aaron shall lay both his hands on the head of the live goat, confess over it all the iniquities of the children of Israel, and all their transgressions, concerning all their sins, putting them on the head of the goat, and shall send it away into the wilderness by the hand of a suitable man. The goat shall bear on itself all their iniquities to an uninhabited land; and he shall release the goat in the wilderness". This is the proverbial “scapegoat”. The prophet Isaiah, in speaking of the Suffering Servant, foretold of the sufferings that Jesus would undergo in the place of His people: “He was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed” (Isa. 53:5). All of Israel’s prophets alluded to the substitutionary nature of the work of the Messiah when they spoke of the work of Salvation. This, of course, also has its foundation in the nature of Old Testament sacrifice.
According to U.S Catholic; The idea of atonement interprets Jesus’ death on the cross as a sacrifice that brings us back into relationship with God. In short, “we are saved because of the cross.” The beginnings of this theology are found in New Testament attempts to find meaning in the execution of Jesus as an enemy of the state and one cursed by God. In the Middle Ages St. Anselm of Canterbury provided the definitive Catholic interpretation of atonement. The question for Anselm, in his work Cur Deus Homo, was how both God’s love and God’s justice can be interpreted in light of the grave reality of human sin. Anselm thought that humankind incurred an infinite debt through original sin and could not pay it back. God desired the flourishing of humankind but could not just allow the results of sin-death. Restitution, or satisfaction, had to be made for the sake of God’s honor and humankind's eternal life. So, God offered God’s own self, through Jesus, as a perfect sacrifice to make payment for the human offense of sin. The crucifixion of Jesus, therefore, sets humankind free to choose eternal life. At its core, atonement is an attempt to help us understand how Jesus’ execution relates to our salvation. It is an attempt to help us understand how we now can be at peace with God despite sin. Yes, Christ died. But he also became human, lived, healed, taught, modeled, and was raised from the dead. And that matters for our salvation too.
The Catechism, while focusing on the redemptive aspects of Christ’s death, gives special attention to Atonement;
615: "For as by one man's disobedience many were made sinners, so by one man's obedience many will be made righteous." By his obedience unto death, Jesus accomplished the substitution of the suffering Servant, who "makes himself an offering for sin", when "he bore the sin of many", and who "shall make many to be accounted righteous", for "he shall bear their iniquities". Jesus atoned for our faults and made satisfaction for our sins to the Father.
623: By his loving obedience to the Father, "unto death, even death on a cross" (Phil 2:8), Jesus fulfills the atoning mission (cf. Is 53:10) of the suffering Servant, who will "make many righteous; and he shall bear their iniquities" (Is 53:11; cf. Rom 5:19)
In essence, is a reconciliatory event. Sin separates us from God, as a barrier that causes estrangement. To atone is to bring that barrier, sin, onto a third being. Jesus brought our sins, our estrangement, onto Himself, thus removing that which separates us from God. This is why many people Jesus saved us from our sins and opened up the gates of Heaven. Sin had barred our entry; Jesus threw open the gates.