A definition would be:
The satisfaction of the righteous demands of God in relation to human sin and its punishment through the sacrificial death of Jesus Christ upon the cross, by which the penalty of sin is cancelled and the anger of God averted. [The NIV is distinctive at this point, in that it generally translates this term by “atonement” and related words.]. Source: Dictionary of Bible Themes, Martin Manser.
An explanation would be:
“Propitiate” and “propitiation” are not commonly used in the English language. We must look to an age long gone in order to discern their meaning. In ancient times, many polytheists thought of their gods as unpredictable beings, liable to become angry with their worshipers for any trifle. When any misfortune occurred, it was believed that a god was angry and was therefore punishing his worshipers. The remedy was to offer the god a sacrifice to appease his anger. This process was called “propitiation.”
A few of the New Testament writers used exactly the same word, but the meaning was slightly different. Instead of seeing God as one whose mood needs to be appeased, “propitiation” focuses on the sacrifice of Jesus by death on the cross which brought the resultant peace between God and sinful humanity.
The Greek term for “propitiation,” hilasmos, occurs in some important passages: Romans 3:25; Hebrews 2:17; 1 John 2:2; 4:10. The message we get from these passages is that propitiation (also called “expiation”) pertains to Christ’s sacrifice for sins in order to bring about a peaceful relationship between God and humanity. Source: Holman treasury of key Bible words, by Carpenter, E. E., & Comfort, P.
The key verses for our word propitiation are in Romans 3:25–26, where Christ Jesus,
whom God put forward as a propitiation by his blood, to be received by faith. This was to show God’s righteousness, because in his divine forbearance he had passed over former sins. 26 It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus.
and in 1 John 2:1-2,
My little children, I am writing these things to you so that you may not sin. But if anyone does sin, we have an advocate with the Father, Jesus Christ the righteous. 2 He is the propitiation for our sins, and not for ours only but also for the sins of the whole world.
The effects of propitiation: God’s justice is satisfied, His wrath has been exhausted, His mercy is available to those who repent.