Hilaskomai [2433]: [The action]

Vine's Notes say that this conveyed the idea of an act done to make the gods propitious or to appease them, because they were viewed as not having good will towards man by nature. This is at odds with the Scriptural view of God, and so the word cannot be seen in this light in Scripture. Rather, it is an act initiated by God, in those places in which the word is used. An interesting quote here: "Never is God said to be reconciled, a fact itself indicative that the enmity [of sin] exists on man's part alone, and that it is man who needs to be reconciled to God, not God to man." So, in essence, the act of propitiation is God's action taken to reconcile man to Himself.

Lk 18:13 The tax-collector begged God to be propitious towards him, recognizing his sinfulness.
Heb 2:17 Christ had to be made every bit man, so that He could be merciful and faithful to make the high-priestly propitiation for our sins.
Heb 8:12 God's promise from the days of the prophets has been that he will be propitious towards man's iniquities and no longer remember their sins, when they turn to Him.

Kapar [OT 3722]: [the action]

indicates the action of covering over - it being unclear whether this means to hide or to wipe away sins. The act of atonement. Appease, pardon, make atonement

Ge 32:20 Jacob sought to appease Esau by the gifts he sent ahead of himself.
Pr 16:14 The wise man seeks to appease the king's wrath.
2Ch 30:18 Hezekiah prayed for the Lord's pardon on his people for not purifying themselves before eating the Passover meal.
Ex 29:36-37 Atonement was made for the altar to consecrate it for seven days before it was made holy.
Ex 30:10 Aaron (the high priest) was to make atonement at the mercy seat once a year.
Ex 30:15-16 All men of age 20+ paid the half-shekel temple tax to make atonement for themselves.
Ex 32:30 Moses sought to make atonement to the Lord for the sins of the people. (But God smote them for the sin of the golden calf.)
Lev 1:4 Hands were laid on the head of the bull to make it an acceptable atonement offering.
Lev 4:20 This same process was done for times when the whole of Israel had erred to make atonement for the nation - leading to forgiveness.
Lev 4:26 And so also for the leader of the nation.
Lev 4:31, 4:35 And for the common people. Always according to ability to pay.
Lev 5:6-13 The same sacrifice applied for those who bore false witness, or who swore thoughtlessly (to do evil or good) in oath.
Lev 5:16 For sins against holy things, too.
Lev 5:18 And even for sins we are unaware of.
Lev 6:7 For robbery,
Lev 6:30 Rules for when the eating of the sacrifice is and isn't acceptable.
Lev 8:15 The beginning of the purification of the altar.
Lev 8:34 Aaron and sons were required to remain in the tabernacle for seven days to complete their ordination, offering daily sacrifices of atonement.
Lev 9:7 Immediately following the period of ordination, Aaron offered a sacrifice of atonement for the sins of himself and the people.
Lev 10:17 Confusion over the eating of the sin offering.
Lev 12:7-8 An offering was to be made after childbirth to atone for the wife, and make her clean again.
Nu 31:50 The officers of Israel's army offered up gold items as an atoning offering in recognition that not a man had been lost in combat.
2Sa 21:3 David needed to make atonement to the Gibeonites for the sins of Saul in breaking his oath to them.
1Ch 6:49 Aaron consecrated the altar and the holy place for Israel.
Ne 10:33 The temple tax was reinstated, but at one-third shekel.
Ez 43:20, 26 Again, the need for consecrating atonement when the new temple was built.
Ez 45:15-20 More rules of atonement - now on a monthly schedule.
Dan 9:24 The sins of Israel had become so great that seventy weeks were now required to make atonement.

Hilasmos [2434]: [the act]

is another word used to signify the means of remittance of sins. It is used of Christ, and indicates that He - in offering Himself up in death - is the sole means by which God shows His mercy to the sinner who believes. Zodhiates seems to distinguish this word as the act, and hilasterion as the place where the act occurs.

1Jn 2:2 Jesus is the means of remittance of our sins, and those of the whole world.
Heb 2:17 True love is seen in God sending Jesus to be this propitiation.

Hilasterion [2435]): [the stage]

(this from Zodhiates) mercy seat, the lid or covering of the Ark of the Covenant - where the blood of the sacrifice was sprinkled on the Day of Atonement, where God promised to meet His people, Christ as the reality of the mercy seat, a place of expiation, an altar or place of sacrifice - not the sacrifices themselves, the symbol of the mercy of the power of God; Vine also notes the reference to the mercy seat, and Christ as the sole means of propitiation. He emphasizes the point that only Christ's blood could be so efficacious in propitiating God, for it was God's own blood given. (He doesn't note the distinction that Zodhiates accentuates.)

Ro 3:25 God showed Christ as the propitiation in His blood through faith to show God's righteousness. [How does this accord with Zodhiates?]
Heb 9:5 The mercy seat was overshadowed by the cherubim of glory.

Kapporet [OT 3727]): [the stage]

Ex 25:17-22 The mercy seat was a gold construct with two golden cherubim attached, such that they faced to seat from its edges, and covered the seat with their outstretched wings. God declared this as the place where He would meet and speak to Moses in giving His commandments to Israel.
Ex 26:34 The mercy seat was put upon the ark of the covenant inside the Holy of Holies.
Ex 30:6 The altar of incense was before the veil just outside [or was it inside? See Lev 16:13-16] the Holy of Holies, at the entrance leading to the mercy seat and the ark.
Ex 35:12 As with all the furnishings of the tabernacle, the mercy seat was to be made by skillful men among the sons of Israel.
Ex 37:6-9 In particular, the seat was made by Bezalel according to God's instructions.
Ex 39:35 The completed work was brought to Moses.
Ex 40:20 The testimony [the tablets? Or does this include the idea of the manna and the rod - both testimonies to God's actions in Israel's history] was kept under the seat.
Lev 16:2 Aaron was not allowed in the Holy of Holies before the mercy seat, because God would appear in the cloud above the seat - and seeing this would be death to Aaron.
Lev 16:13-15 On the Day of Atonement, Aaron would enter in, but only after putting incense on the altar so the cloud could cover the seat, and protect him from seeing what he ought not to see. Within that cloud, he was to sprinkle the blood of the bull, followed by the blood of the goat. This was to make atonement for the tabernacle itself, which had become impure due to its being amidst the impure sons of Israel.
Nu 7:89 God spoke to Moses from above the mercy seat.

Summary

So then, the action of propitiation is an action of God towards man. Men have often sought to appease God, but in reality, it has been God helping man to reunite with Him. The details in Leviticus show us a great deal about this action. It was always done by transference - hands laid upon the sacrificial offering as an identifying of our guilt with its death (Lev 1:4). By faith, this remains true today, as we identify our guilt with the death of the Son of God. The offering required was never beyond the guilty party's ability to pay. (Lev 4:31, 4:35, etc.). As Christ is our offering, His perfect sacrifice is now within everybody's ability to pay. What remains now is the choice as to whether we will pay. The reasons we find ourselves needing to seek propitiation are manifold: personal sins - either in service or in daily life, the sins of the nation or its leaders, misuse of the holy, purification of things dedicated to holiness, unknown sins, etc. Lk 18:13 makes it evident that we seek God to be propitious towards us - to reunite us in our enmity. Heb 2:17 shows us that Christ has assumed the role of high priest, offering Himself as the propitiatory sacrifice. Heb 8:12 tells us that this was foretold throughout the Old Testament, Christ fulfilling the types of the earlier covenant.

The act itself was accomplished in Jesus. He was sent of God as a sign of God's love for man (1Jn 4:10), offering Himself sinless and yet willing. In His death, the propitiation for all of the sins of the world was accomplished (1Jn 2:2). It remains for us to accept the payment offered on our behalf.

The stage is ever the mercy seat, of which again Christ is the fulfilling of the type. This was - in both type and fulfillment - according to God's design (Ex 32:6-9, 1Jn 4:10). It was the cap upon the testimony of God's action in history (Ex 40:20), and the seat from which God spoke to His people (Nu 7:89). Both of these aspects are also fulfilled by Christ, who was and will be the crown of God's actions in the history of mankind, and who spoke only what He heard from the Father while on earth. It signifies the place where God appears to man, revealing Himself. Aaron could not approach without filling the Holy of Holies with incense (Lev 16:2, Lev 16:13-15). Today, we can approach Him, but only in the death of our fleshly ways. It is greatly interesting to realize that atonement was necessary for the tabernacle, as it sat amidst sinful Israel. Today, we are that tabernacle, and we too sit amidst a sinful people - even housing the tabernacle in bodies of sinful flesh. How much more, then, do we need the constant atoning offered us in Christ who fills our temple?