Propitiation, also called by some expiation, is the act of appeasing or making well-disposed a deity, thus incurring divine favor or avoiding divine retribution.
In Romans 3:25 the NASB translates "propitiation" from the Greek word hilasterion. Concretely it specifically means the lid of The Ark of The Covenant. The only other occurrence of hilasterion in the NT is in Hebrews 9:5, where the NASB translates it as "mercy seat".
For many Christians it has the meaning of "that which expiates or propitiates" or "the gift which procures propitiation". (KJV) reads: "And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world." There is frequent similar use of hilasterion in the Septuagint, ff. The mercy seat was sprinkled with blood on Yom Kippur , representing that the righteous sentence of the Law had been executed, changing a judgment seat into a mercy seat (; compare with "throne of grace" in ; place of communion, .
Another Greek word, hilasmos, is used for Christ as our propitiation in 1 John 2:2; 4:10; and in the Septuagint (; ; ). The thought in the OT sacrifices and in the NT fulfillment, is that Christ completely satisfied the just demands of the Holy Father for judgment on sin, by his death at Calvary .TDNT, however, takes a different view of Hebrews: "If the author uses the ritual as a means to portray Christ's work, he also finds that in the new covenant the literal offerings of the ritual are replaced by the obedience of Christ (10:5ff.; cf. Ps. 40) and the Christian ministry of praise and mutual service (13:15-16; cf. Ps. 50). In other words, total self-giving, first that of Christ, and then, on this basis, that of his people, is the true meaning of sacrifice.
God, in view of the cross, is declared righteous in having been able to justify sins in the OT period, as well as in being able to forgive sinners under the New Covenant (; cf. , note). 
The case for translating hilasterion as "expiation" instead of "propitiation" was put forward by C. H. Dodd in 1935 and at first gained wide support. Hilasterion is translated as "expiation" in the New American Bible (Revised Edition) and as "sacrifice of atonement" in the New Revised Standard Version. Dodd argued that in pagan Greek the translation of hilasterion was indeed to propitiate, but that in the Septuagint (the oldest Greek translation of the Hebrew OT) that kapporeth (Hebrew for "covering") is often translated with words that mean "to cleanse or remove". This view was initially challenged by Roger Nicole in twenty-one arguments.Later it was also challenged by Leon Morris who argued that because of the focus in the book of Romans on God's wrath, that the concept of hilasterion needed to include the appeasement of God's wrath. In his semantic study of hilasterion David Hill claims that Dodd leaves out several Septuagint references to propitiation, and cites apocryphal sources.
Theologians stress the idea of propitiation because it specifically addresses dealing with God's wrath. Critics of penal substitutionary atonement state that seeing the Atonement as appeasing God is a pagan idea that makes God seem tyrannical.
J.I. Packer in "Knowing God" designates a distinct difference between pagan and Christian propitiation: "In paganism, man propitiates his gods, and religion becomes a form of commercialism and, indeed, of bribery. In Christianity, however, God propitiates his wrath by his own action. He set forth Jesus Christ... to be the propitiation of our sins."
John Stott writes that propitiation "does not make God gracious...God does not love us because Christ died for us, Christ died for us because God loves us". John Calvin, quoting Augustine from John's Gospel cx.6, writes, "Our being reconciled by the death of Christ must not be understood as if the Son reconciled us, in order that the Father, then hating, might begin to love us". Continuing the quote: "... but that we were reconciled to him already, loving, though at enmity with us because of sin. To the truth of both propositions we have the attestation of the Apostle, 'God commendeth his love toward us, in that while we were yet sinners, Christ died for us,' (Rom. 5: 8.) Therefore, he had this love towards us even when, exercising enmity towards him, we were the workers of iniquity. Accordingly, in a manner wondrous and divine, he loved even when he hated us." 
Packer also cites God's love as the impetus that provides Christ's sacrifice for the reconciliation of mankind and hence the removal of God's wrath. According to Packer, propitiation (and the wrath of God that propitiation implies) is necessary to properly define God's love; God could not be righteous and "His love would degenerate into sentimentality (without Christ's death containing aspects of propitiation).The wrath of God is as personal, and as potent, as his Love."
Thus the definition of Christian propitiation asserted by Calvin, Packer and Murray holds that within God there is a dichotomy of love and anger, but through propitiation love trumps anger, abolishing it. "'The doctrine of the propitiation is precisely this that God loved the objects of His wrath so much that He gave His own Son to the end that He by His blood should make provision for the removal of this wrath... (John Murray, The Atonement, p.15)'"
The New Jerome Biblical Commentary argues that in the NT sacrifice (hilasterion) does not appease God's wrath but is best expressed from its Jewish roots (76.89-95) as atonement or expiation (82.73). Recent Catholic studies have depended heavily on the Trinitarian perspective presented by Edward J. Kilmartin:
Sacrifice is not, in the first place, an activity of human beings directed to God and, in the second place, something that reaches its goal in the response of divine acceptance and bestowal of divine blessing on the cultic community. Rather, sacrifice in the New Testament understanding - and thus in its Christian understanding - is, in the first place, the self-offering of the Father in the gift of his Son, and in the second place the unique response of the Son in his humanity to the Father, and in the third place, the self-offering of believers in union with Christ by which they share in his covenant relationship with the Father.
Robert Daly has explained the background for this renewed understanding. Daly points out that the initiative is entirely with the Father who "loved us and sent his Son as expiation for our sins" (1 John 4:10), and that "when we see the sacrifice of Christ and the sacrifice of the Mass as a Trinitarian event, we see that, strictly speaking, there are no recipients." He compares the Eucharist to a marriage ceremony that receives its meaning by becoming the reality of one's life.
In line with this, the entry on "sacrifice" in the Theological Dictionary of the New Testament, after reviewing the epistles of Paul and Hebrews, concludes that "total self-giving, first that of Christ, and then, on this basis, that of his people, is the true meaning of sacrifice." And Cardinal-theologian Walter Kasper, in his book The God of Jesus Christ, concludes that what Jesus effected was to give suffering "eternal import, the import of love." Kasper points out that Gregory of Nyssa and Augustine of Hippo, working out of the New Testament, speak of a God who can freely choose to feel compassion, which implies suffering. Kasper adds that: "It is Origen who gave us the clearest statement. In Origen's words: 'First God suffered, then he came down. What was the suffering he accepted for us? The suffering of love.' Origen adds that it is not just the Son but also the Father who suffers so. This is made possible by God's freedom in love."