appease, make (an atonement, cleanse, disannul, forgive, be merciful, pacify, pardon, purge (away), put off, (make) reconcile(-liation)

Boyd’s Bible Dictionary:

(to cover). First, a work presented to God that perfectly glorifies His name and character about sin by sacrifice, and second, the bearing of our sins. By the blood-shedding of Christ, the believer may have the certainty that his sins are all gone, that God will remember them no more (Ex. 30:16; Lev. 1; Lev. 14-17; Lev. 23:27-32).

“Forgiveness” From Concise Bible Dictionary:

There are three Hebrew words translated to forgive.
1. kaphar, “to cover” (Deut. 21:8; Psa. 78:38; Jer. 18:23). It is also translated “atonement.”
2. nasa, “to bear,” take away [guilt]: used by Joseph's brethren when they asked him to forgive them (Gen. 50:17); and used of God as "forgiving iniquity and transgression and sin" (Ex. 34:7; Num. 14:18); and in describing the blessedness of the man "whose transgression is forgiven, whose sin is covered" (Psa. 32:1).
3. salach, “to pardon,” used only of the forgiveness that God gives. It is employed for the forgiveness attached to the sacrifices: "it shall be forgiven him" (Lev. 4:20, 26, 31, 35; Lev. 5:10, 13, 16, 18; &c). It occurs in the prayer of Solomon at the dedication of the temple (1 Kings 8:30, 34, 36, 39, 50. Also in Psa. 103:3; Jer. 31:34; Jer. 36:3; Dan. 9:19).
In the N. T. two words are used: ὔφεσις., from ἀφίημι, “to send from, release, remit,” several times translated REMISSION; and χαριξομαι, “to be gracious, bestow freely, forgive.” Both words are applied to the forgiveness granted by God, as well as that between man and his fellow.
There are two aspects in which forgiveness is brought before us in scripture.
1. The mind and thought of God Himself towards the sinner whom He forgives. On the ground of the sacrifice of Christ, God not only ceases to hold those who have faith in Christ's blood as guilty before Him, but His favor is towards them. "Their sins and iniquities will I remember no more" (Heb. 10:17). Thus all sense of imputation of guilt is gone from the mind of God. "God for Christ's sake hath forgiven you" ἐχαρίσυτο, graciously forgiven, (Eph. 4:32). So in the O.T., "I will heal their backsliding, I will love them freely" (Hos. 14:4).
2. The guilty one is released, forgiven. "That they may receive forgiveness of sins" (Acts 26:18). "As far as the east is from the west, so far hath he removed our transgressions from us" (Psa. 103:12). "Your sins are forgiven you for his name's sake" (1 John 2:12). Hence it is true of all Christians, that their sins are forgiven. Another thought is included in the forgiveness of sins, namely, that having redemption by Christ, which brings into a new state, the whole guilty past is forgiven, removed from us, so that there is no hindrance to the enjoyment of that into which redemption brings.
The general principle as to forgiveness is stated in 1 John 1:9: "If we confess our sins, he is faithful and just to forgive us our sins;" and to this is added, "and to cleanse us from all unrighteousness." This involves honesty of heart, whether in a sinner first coming to God, or in a child who has grieved the heart of the Father by sinning. The two aspects above referred to are here also. The faithfulness and righteousness of God in forgiving, and the cleansing us from all unrighteousness. God is faithful to His own blessed character of grace revealed in His Son, and righteous through the propitiation which He has made.
3. If a Christian is “put away” from the assembly and is repentant, he is forgiven and restored (2 Cor. 2:7, 10). This of course is different from the act of God in forgiving sins, and may be called administrative forgiveness in the church; and if the act of discipline is led of the Spirit, it is ratified in heaven: (Compare John 20:22-23). This is entirely different from any pretended absolution that may be pronounced over poor deluded unconverted persons.
4. There is also a governmental forgiveness in connection with the government of God here below in time, both on God's part, and toward one another (Isa. 40:1-2; Luke 17:3; James 5:15-16; 1 John 5:16). We are called upon to forgive one another; and if we indulge in a harsh unforgiving spirit, we must not expect our Father to forgive us in His governmental dealings (Matt. 6:14-15).

“Reconciliation” From Concise Bible Dictionary:

Except in 1 Samuel 29:4, and 2 Chronicles 29:24, the Hebrew word is kaphar, which is more than sixty times translated “to make an atonement;” and this rendering suits sufficiently well in the places where “reconciliation” is read in the AV (Lev. 6:30; Lev. 8:15; Lev. 16:20; Ezek. 45:15,17,20; Dan. 9:24). In the New Testament the last clause of Hebrews 2:17 should be translated “to make ‘propitiation’ for the sins of the people.” Elsewhere the word translated “reconciliation” is καταλλαγἠ, and kindred words, signifying “a thorough change.”
By the death of the Lord Jesus on the cross, God annulled in grace the distance which sin had brought in between Himself and man, in order that all things might, through Christ, be presented agreeably to Himself. Believers are already reconciled, through Christ’s death, to be presented holy, unblameable, and unreproveable (a new creation). God was in Christ, when Christ was on earth, reconciling the world unto Himself, not imputing unto them their trespasses; but now that the love of God has been fully revealed in the cross, the testimony has gone out worldwide, beseeching men to be reconciled to God (2 Cor. 5:19-20). The end is that God may have His pleasure in man.
Christ also abolished the system of the law that Jew and Gentile might be reconciled together unto God, the two being formed in Christ into one new man (Eph. 2:15-16). Reconciliation will extend in result to all things in heaven and on earth (Col. 1:20); not to things under the earth (the lost), though these will have to confess that “Jesus Christ is Lord, to the glory of God the Father” (Phil. 2:10-11).

“Atonement, Day Of” From Concise Bible Dictionary:

This was one of the most solemn days in the whole year, being, in common with the Sabbath, the only occasion on which the people were commanded to cease from work entirely. On the day of atonement they were also to afflict their souls, and that by a statute forever (Lev. 16:29-31). The time of the year in which the day fell — the tenth day of the seventh month — is very significant, especially when viewed in connection with the other feasts. See FEASTS.
The rites prescribed for the Day of Atonement are given (Lev. 16; Lev. 23:26-32; Num. 29:7-11). In the first we have a detailed account of the peculiar work appointed for the high priest on that day; in the second, we learn what had been shortly expressed in Leviticus 17, how the people should comport themselves on that day; and in the third we are told of certain sacrifices which were to be offered up besides those spoken of in Leviticus 16. In this passage we learn that the Day of Atonement was a gracious provision in order that the relationship of the people with the holy God who dwelt among them might be maintained.
The points to be noticed are
1. The entrance into the holy place. Aaron, on account of the failure of his sons in the priestly office, could not enter there at all times, but, as the representative of the people, once a year on the Day of Atonement, and he must enter alone. Even then he went in not for communion, but for the cleansing of the defilements of a people among whom God dwelt. He must put on the holy linen garments, and must enter with a cloud of incense and with blood, lest he die. For the contrast to this for the Christian (see Heb. 9:7-12).
2. Aaron must offer sacrifices for himself and his house; a young bullock for a sin offering and a ram for a burnt offering. Aaron and his sons represent the saints who now form the church as a company of priests, and were thus, in the type, distinct from the people (Israel) as an earthly company who formed the camp.
3. For the congregation two goats were taken for a sin offering, and a ram for a burnt offering. On the two goats the lot was cast, and the one on whom the lot fell was for Jehovah, and was offered as a sin offering. This, as with the bullock for Aaron and his house, was the atonement offering Godward. The other, after being presented before the Lord, was brought forth: on him Aaron laid both his hands and confessed over him “all the iniquities of the children of Israel and all their transgressions in all their sins, putting them upon the head of the goat,” (Lev. 16:21), which was then sent away into the wilderness, a land of forgetfulness. In the two goats we have the two sides of atonement, namely, that which meets the character and holiness of God, and that which meets the need of the sinner as to the removal of his sins. (See ATONEMENT.)
4. Atonement was made for the holy place, for the tabernacle, and for the altar, because of the uncleanness of the children of Israel.
The day of Atonement being once a year — once every year, by a perpetual statute — stands in strong contrast to the one perfect sacrifice of the Lord Jesus, by which the believer is perfected in perpetuity. (See Heb. 10:1-18).
The reiterated injunction to afflict their souls in connection with this great day has an important prophetical bearing as to the Lord’s earthly people Israel. If the blowing of trumpets on the first day of the seventh month suggests that testimony from the Lord which shall arouse the nation from their long spiritual slumber, the Day of Atonement points to the moral effect produced in them when they shall look upon Him whom they have pierced, and mourn for Him. They will also at this time learn and confess that He was wounded for their transgressions (Zech. 12:10-14; Isa. 53; Psa. 51).

Strong’s Dictionary of Hebrew Words:

Transliteration:
kaphar
Phonic:
kaw-far’
Meaning:
a primitive root; to cover (specifically with bitumen); figuratively, to expiate or condone, to placate or cancel
KJV Usage:
appease, make (an atonement, cleanse, disannul, forgive, be merciful, pacify, pardon, purge (away), put off, (make) reconcile(-liation)