To understand what a propitiation is it is helpful to understand what it is not. This is often a surprise to people but the NT never speaks of Jesus' death as an "atonement". That is because an "atonement" is an expression of contrition and an appeal for forgiveness made by or on behalf of the perp (the one who sinned). Jesus was not making atonement.
In contrast, for Paul in the context of Romans 3 at least, a propitiation is made by a judge to appease the populace. That it is God and not Jesus who was making propitiation can be seen in the context (my translation):
Rom 3:25 - 26 God hath set forth Jesus as a propitiation by means of
the good faith he expressed in the shedding of Jesus' blood, publicly
demonstrating God's uprightness in the matter of the remission of sins
done in the past through God's merciful disposition; To declare, I
say, in the present God's uprightness in order that he might be
justified, and the justifier of the one that puts faith in Jesus.
That's still a little dense so let me explain what's going on...
First of all we have to appreciate the fact that as judge of all the earth, God has an obligation to the populace to avenge harm done:
Rom 12:19 Dearly beloved, don't take revenge yourselves, but rather
give leave it to God: because it is written, "Vengeance is my
responsibility which I will perform," says the Lord.
So while on a personal level God is completely free to forgive the sins of anyone who has harmed himself but as judge he must not be remiss by not avenging those who have harmed others. And yet he did. When the Jews came to God in contrition and sought his forgiveness he freely forgave them because of his merciful disposition. So it was necessary to appease those who had given place to wrath in the confidence that God would one day avenge them and yet God had opted to forgive them.
It is for this that God had to show good faith by being among those who suffered wrong and yet forgave and did not get their revenge. He did this by setting forth Jesus before the populace, suffering at the hands of people that he would later forgive and never extract revenge.
By doing this he is cleared of reproach for freely forgiving sins in the past and also for forgiving the sins of those who have faith in Christ in the present and future. It is not Jesus who takes away sin but God. God forgives freely on the basis of faith, not sacrifice. The death of Jesus was to justify God for doing so.
It may be an uncomfortable thought to you that God made propitiation to the populace to appease them and clear himself but that is precisely what Paul says. Please re-read it slowly and you'll have to agree.
John refers to the same thing here:
1Jn_1:9 If we confess our sins, he is faithful and just to forgive us
our sins, and to cleanse us from all unrighteousness.
It is the propitiation that makes him faithful and justified.
1 John may or may not be using the term in the same way.
A good illustration of this usage of ἱλαστήριον can be found in 4 Maccabees 17 wherein a demonstration, the valiant death of a mother and her seven children saved the Jews from Antiochus who saw their courage and uprightness and turned from the Jews to his other enemies. Here's the salient passage:
4Ma 17:20 These, therefore, having been sanctified through God, have
been honoured not only with this honour, but that also by their means
the enemy did not overcome our nation; 4Ma 17:21 and that the tyrant
was punished, and their country purified. 4Ma 17:22 For they became
the atnipoised to the sin of the nation; and the Divine Providence
saved Israel, aforetime afflicted, by the blood of those pious ones,
and the propitiatory death. 4Ma 17:23 For the tyrant Antiochus, looking to their manly virtue, and to their endurance in torture,
proclaimed that endurance as an example to his soldiers. 4Ma 17:24
And they proved to be to him noble and brave for land battles and for
sieges; and he conquered and stormed the towns of all his enemies.