Penal-substitutionary theory (including the concept of propitiation and expiation): God's wrath is dealt with by propitiation; sins are cleansed.
Ransom theory deals with one's bondage to sin or evil. Jesus's death is a ransom, and the price is paid to the devil. As a result, sinners are released from bondage.
Satisfaction theory: Jesus's death is a sinless sacrifice that satisfies God's justice.
But the traditional atonement theories do not consider seriously the question of why Jesus was put to death in a historical, political sense. According to the Gospels and Paul's major letters, Jesus's death has to do with his radical message of God's rule in the here and now.
The challenge for us is interpreting various passages regarding Jesus's death in the New Testament. The following translations are from the NRSVue.
Mark 14:24: “This is my blood of the covenant, which is poured out for many."
1 Cor 5:7: "Clean out the old yeast so that you may be a new batch of dough, as you really are unleavened. For our paschal lamb, Christ, has been sacrificed."
Matt 26:28: "for this is my blood of the covenant, which is poured out for many for the forgiveness of sins."
c.f. Matt 8:17: "This was to fulfill what had been spoken through the prophet Isaiah, 'He took our infirmities and bore our diseases.'"
2 Cor 5:21: "For our sake God made the one who knew no sin to be sin, so that in him we might become the righteousness of God."
Rom 3:25: "whom God put forward as a sacrifice of atonement (hilasterion) by his blood, effective through faith. He did this to demonstrate his righteousness, because in his divine forbearance he had passed over the sins previously committed;"
*Note: Hilasterion may be translated or interpreted variously; it may be a propitiation, sacrifice of atonement, or mercy seat. In the Septuagint, hilasterion is the translation of the Hebrew Kaporet, which means the cover of the ark of the covenant on the Day of Atonement (Yom Kippur).
Rom 5:6-10: "For while we were still weak, at the right time Christ died for the ungodly. 7 Indeed, rarely will anyone die for a righteous person—though perhaps for a good person, someone might actually dare to die. 8 But God proves his love for us in that while we still were sinners Christ died for us. 9 Much more surely, therefore, since we have now been justified by his blood, will we be saved through him from the wrath of God. 10 For if while we were enemies we were reconciled to God through the death of his Son, much more surely, having been reconciled, will we be saved by his life."
Gal 1:4: "Who gave himself for our sins to set us free from the present evil age, according to the will of our God and Father,"
Heb 10:12: "But when Christ had offered for all time a single sacrifice for sins, 'he sat down at the right hand of God,'"
1 John 2:2: "And he is the atoning sacrifice for our sins, and not for ours only but also for the sins of the whole world." *atoning sacrifice (hilasmos)
1 John 4:10: "In this is love, not that we loved God but that he loved us and sent his Son to be the atoning sacrifice for our sins." *atoning sacrifice (hilasmos)
1 Pet 2:23-24: "When he was abused, he did not return abuse; when he suffered, he did not threaten, but he entrusted himself to the one who judges justly. 24 He himself bore our sins in his body on the cross, so that, having died to sins, we might live for righteousness; by his wounds you have been healed."