I believe He died for the sin of the world caused by Adam and Eve that separated us from God's Holy Spirit ,by His sacrificial death He brought God's Holy Spirit back to us for our conviction that leads those who believe Him that sent Him to all righteousness.
Atonement is the process of forgiving or pardoning a transgression
past,present and future by conviction,confession and repentance that leads us to all righteousness through Christ.
We must be reconciled by the Father through His Son before we receive the reconciliation which is the body and mind of Christ (we are reconciled by Him and for Him) this is the atonement process.
The reason for our Lord Jesus sacrificial death was to overcome the world which was the sin of the world that occurred from the beginning. That sin upheld the atonement process from occurring a new heart and mind.
When our Lord died the veil was torn in two and the Holy Spirit came to it's rightful resting place (temple of man) where the reasoning of all hearts would be seen by the Father's Holy Spirit through our Lord Jesus Christ and the atonement process by the conviction of the Holy Spirit now takes place as we receive a new heart and have the law inscribed in our minds.
By the conviction of the Holy Spirit it leads us to all righteousness through our Lord King and Saviour Jesus Christ by the hand of our Father.
Is it nothing to all you who pass this way? Look and see if there is any pain like my pain Which was severely dealt out to me, Which the LORD inflicted on the day of His fierce anger. Lamentations 1:12
"But He answered and said, 'I was sent only to the lost sheep of the house of Israel.'" Matthew 15:24
"and not for the nation only, but in order that He might also gather together into one the children of God who are scattered abroad." John 10:16
"I have other sheep, which are not of this fold; I must bring them also, and they will hear My voice; and they will become one flock with one shepherd." John 10:16
Atonement merely covered sins but could do nothing to take them away (i.e. blood of bulls)
From kaphar; properly, a cover, i.e. (literally) a village (as covered in); (specifically) bitumen (as used for coating), and the henna plant (as used for dyeing); figuratively, a redemption-price -- bribe, camphire, pitch, ransom, satisfaction, sum of money, village.
"And according to the Law, one may almost say, all things are cleansed with blood, and without shedding of blood there is no forgiveness." Hebrews 9:22
Animal sacrifices atoned for sins, or 'covered over,' but only Christ could make propitiation, or take them away and satisfy God's Justice
"For it is impossible for the blood of bulls and goats to take away sins.
Therefore, when He comes into the world, He says, "SACRIFICE AND OFFERING YOU HAVE NOT DESIRED, BUT A BODY YOU HAVE PREPARED FOR ME; IN WHOLE BURNT OFFERINGS AND sacrifices FOR SIN YOU HAVE TAKEN NO PLEASURE. "THEN I SAID, 'BEHOLD, I HAVE COME (IN THE SCROLL OF THE BOOK IT IS WRITTEN OF ME) TO DO YOUR WILL, O GOD.'" After saying above, "SACRIFICES AND OFFERINGS AND WHOLE BURNT OFFERINGS AND sacrifices FOR SIN YOU HAVE NOT DESIRED, NOR HAVE YOU TAKEN PLEASURE in them" (which are offered according to the Law), then He said, "BEHOLD, I HAVE COME TO DO YOUR WILL." He takes away the first in order to establish the second. By this will we have been sanctified through the offering of the body of Jesus Christonce for all.Every priest stands daily ministering and offering time after time the same sacrifices, which can never take away sins; but He, having offered one sacrifice for sinsfor all time, SAT DOWN AT THE RIGHT HAND OF GOD, waiting from that time onward UNTIL HIS ENEMIES BE MADE A FOOTSTOOL FOR HIS FEET. For by one offering He has perfected for all time those who are sanctified." Hebrew 10:4-14
"Therefore the Law has become our tutor to lead us to Christ, so that we may be justified by faith." Galatians 3:24
But propitiation implies a taking away of sins and a full satisfaction of God's Wrath and His justice, which Christ did by His Sacrifice on the Cross for the sins of His people...
"As a result of the anguish of His soul, He will see it and be satisfied; By His knowledge the Righteous One, My Servant, will justify the many, As He will bear their iniquities." Isaiah 53:11
"In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins." I John 4:10
"whom God displayed publicly as a propitiation in His blood through faith. This was to demonstrate His righteousness, because in the forbearance of God He passed over the sins previously committed;" Romans 3:25
"In Him we have redemption through His blood, the forgiveness of our trespasses, according to the riches of His grace" Ephesians 1:7
"Therefore, He had to be made like His brethren in all things, so that He might become a merciful and faithful high priest in things pertaining to God, to make propitiation for the sins of the people." Hebrews 2:17
atonement [ Middle English atonen (to be reconciled; to restore oneness or unity) < at (at) + one (one) ] Amends made for a wrong, sin, or crime. Reconciliation between parties who are seriously at odds, especially between created beings and the Deity. Or, an instance or cause of such a reconciliation. Related words include expiation, repayment, amends, satisfaction, and propitiation.
expiation: [ < Latin expiare, < ex- ('out' - here used as an intensifier) + piare (to atone) < pius (devout)] to compensate, repay, make amends for, make up for.
propitiation [ < Latin propitius (favorable, auspicious), < pro- (forward) + -pitius, (to eagerly volunteer or give of oneself) -- lit., rushing forward (to the altar or throne) to please the angry god or ruler ] : To appease, pacify, placate, assuage, or mollify. Propitiation is sometimes used to translate Greek hilaskesthai.
Of course, our God is the Only God, not to be compared with the dung-hill gods of the idolatrous nations, He can not be 'bought-off,' or bribed. He is ultimately only satisfied with the exclusive Sacrifice of His Son, the Only Perfect Sacrifice that only He could provide to save men from the penalty of their sin. Only the Righteousness of God can satisfy the Righteous demands of His Perfect Law. Our sin was the rebellion of the finite creation against the Almighty Holy Creator. Out of Love and for His Glory God sent His Son, who went foreward to His death, carrying His Cross, and bearing the sins of the world (not just the sins of the Jewish nation). Only His Sacrifical death on the cross justifies once ungodly, sinners and hides them in Christ, covers them in a Robe of His Righteousness and Garments of Salvation. Only by being born again of the Holy Spirit may one dead in their sins be given new life as it is in Christ, and empowered for Holy Living.
Only the High Priestly Work of Christ sanctifying the prayers and offerings of His people makes them presentable and acceptable before God as a sweet fragrance of Christ.
"being justified as a gift by His grace through the redemption which is in Christ Jesus;" Romans 3:24
"Since we have now been justified by his blood, how much more shall we be saved from God's wrath through him!" Romans 5:9
"Therefore, having been justified by faith, we have peace with God through our Lord Jesus Christ," Romans 5:1
Today, few people know what 'propitiation' means, so the word 'atonement' has largely been used instead of propitiation.
Animal sacrifices atoned for sins, Christ took them away by becoming the propitiation for our sins.
In today's language, you might say
He died to atone as the Perfcet Sacrifice for our sins, taking them away and nailing them to His Cross, with His blood He bought my freedom and quenched the wrath of God on my behalf, thank You Jesus