The earliest gospel, Mark, has only one explicit statement on the significance of Christ's death: Mark 10:45. "the Son of Man did not come to be served, but to serve, and to give his life as a ransom for many."  Matthew and Luke have little more, and Luke even softens this verse.  The temple curtain does rip from top to bottom (e.g., Mark 15:38), which may imply that Jesus' death was a final atonement for sins or that those who trust in his death now have direct access to God. But the Synoptic Gospels do not say what the significance is.
As we would expect, the Gospel of John is more explicit. "Whoever eats my flesh and drinks my blood has eternal life" (John 6:54). So John tells us that Jesus' death is the mechanism by which we can have eternal life. John doesn't specify exactly how it works, but we have good reason to think in sacrificial terms. For example, John seems to have moved Jesus' action in the temple (2:13-25) next to Jesus turning water into wine (2:1-12). The two stories go hand in hand.
The jars into which Jesus has them put water are for the waters of ritual purification (2:6). Turning water into wine may very well then symbolize the transition from Old Testament purification by water to the "wine" of Jesus' blood. Similarly, Jesus "destroys" the temple with his death and raises it up anew when he rises from the dead.
In the minds of the ancients, sacrifices functioned on a deep level. Perhaps most basically, they kept the gods happy, well fed, as it were. Christian thinkers have long tried to unpack the significance of Christ's death, and the views that relate most to Jesus' death as a sacrifice are the views that see his death as a satisfaction of God's anger toward sin or even some basic satisfaction of the need for justice. Paul puts it this way, "God presented Christ as a sacrifice of atonement, through the shedding of his blood—to be received by faith. He did this to demonstrate his righteousness, because in his forbearance he had left the sins committed beforehand unpunished" (Rom. 3:25).
Jesus's death shows that God is just, even though he is choosing not to hold us accountable for our past sins. There is a general sense, then, that Jesus took our place. He died "for our sins" (1 Cor. 15:3). Mind you, some have taken this picture of Jesus' death far beyond anything the New Testament says. Paul doesn't treat Christ's death as some mathematical equivalent for the exact amount of punishment our sins demand from a standpoint of justice.  "For our sins" is the reason he died, not a statement that he substituted for our exact punishment.
Jesus' death satisfies the order of things. It powerfully demonstrates what God stands for. His grace to us came at a huge price. It also demonstrates how much he loves us, so much that the Father is willing to sacrifice his own life for us. Indeed, Jesus himself is willing to die for us.
Christianity has never endorsed a single view of how Jesus' death worked. It was a sacrifice. It satisfied the order of things. Jesus took our place. Jesus showed us God's love and his obedience to death is a model for us to follow. All these pictures of Jesus' death are true.
A final one is the idea that in his sinless death, Jesus defeated the powers of Satan. The image of a "ransom" relates somewhat to this view, as well as to the satisfaction view. Jesus' death "pays off" the price to free humanity from sin and death. It is a metaphor that shouldn't push it too far. The "Christus Victor" view is the one that looks at Christ's death primarily as the defeat of Satan and the defeat of the evil powers that enslave the world.
However it might exactly work, the effect of Jesus' death is reconciliation to those who trust in it. God has "reconciled us to himself through Christ" (2 Cor. 5:18). We are "buried with him through baptism into death" (Rom. 6:4). We "have been crucified with Christ" (Gal. 2:20). We die with him and will thus rise as he did as well.
All this reflection arguably came after Jesus' time on earth. Jesus' death played almost no role at all in Jesus' own early message and explicit mission. Perhaps only in his very last days did he begins to speak to his disciples of his death as a kind of sacrifice.
This observation does not in any way deny the central and essential importance of his death. However, sometimes it seems like some Christians ignore the bulk of Jesus' message while he was on earth and focus disproportionately on the saving significance of his death and resurrection. We can preach the good news of Jesus' death and resurrection without ignoring the good news of the kingdom he brought while he ministered in Galilee.
 We already discussed earlier in the chapter the probable significance of Jesus' words at his Last Supper.
 Luke 19:10 simply says "The Son of Man came to seek and to save the lost," leaving out the sense of Jesus' death as a ransom. Luke has nothing to say about the atoning significance of Christ's death but is instead focused much more on the significance of his resurrection.
 A view sometimes called penal satisfaction.