The details for this day's original observance are recorded in Leviticus chapter 16, as given by God through Moses to the Israelites in the wilderness after their miraculous escape from Egyptian slavery and God's covenant-making with them at Mt. Sinai. Everything about this day emphasized the holiness and inaccessibility of God as well as his mercy and grace.
On this day alone the High Priest put on white linen clothes in place of his bejeweled ceremonial vestments and entered the Holy of Holies, the inner sanctum of the Tabernacle and later of the Temple. With him, he brought the blood of sacrificial animals, offered first for his own sins and then for those of the people, which blood he sprinkled on the Mercy Seat above the Ark of the Covenant. He came into God's ceremonial presence behind a cloud of incense smoke, "lest he die."
Yom Kippur involved two goats. One was sacrificed for sin and its blood sprinkled as described above. On the head of the second animal, called the Scapegoat, the High Priest placed his hands and confessed all the sins of the people of Israel, symbolically transferring those sins on to this goat. An appointed man then led that goat into the wilderness, carrying those sins, where it was abandoned. The High Priest took off his linen clothes, bathed his body, redressed in his usual ceremonial vestments and came out to the people.
The word for the Mercy Seat in the Greek Old Testament was hilasterion, the "Propitiation," a word combining the ideas of appeasement and reconciliation. By offering himself, symbolized in the shedding of his blood on the Cross, Jesus accomplished propitiation for all who will ever be saved, whether they lived before Jesus or since. Since Jesus had no sins of his own, his self-sacrifice was effective for all time and needed offering only once in human history.
As the Scapegoat symbolically carried Israel's sins into the wilderness, Jesus removed forever the guilt of sin from all those whom he represented. As the High Priest went into the Most Holy Place into God's symbolic presence, Jesus rose from death then ascended into heaven into the actual presence of God where he intercedes for his people. As the High Priest finally came out from the Most Holy Place to the forgiven people, Jesus will return some day from heaven to receive all his waiting people, whether still living or raised back from death.
Those Jews who did not receive the Messiah were devastated and felt obligated to re-invent Judaism for a situation without any Temple or priests or sacrificial animals. Over time, repentance and prayer were said to have replaced the Old Testament externals and so it remains in Judaism today.
For all those who will see it, whether Jew or Gentile, the great Day of Atonement occurred nearly 2,000 years ago on Golgotha. There God's wrath against sin was poured out completely, his mercy toward sinners was displayed in public view, and access between the holy God and his wayward but repentant creatures was procured and certified for all time and for eternity. Blessed be GOD, who forgives all our sins!
For a detailed study of the Epistle to the Hebrews, in which the unknown author presents Jesus' death as the fulfillment of the meaning of the Day of Atonement, click here.