אַךְ בֶּעָשׂוֹר לַחֹדֶשׁ הַשְּׁבִיעִי הַזֶּה יוֹם הַכִּפֻּרִים הוּא, מִקְרָא-קֹדֶשׁ יִהְיֶה לָכֶם, וְעִנִּיתֶם, אֶת-נַפְשֹׁתֵיכֶם; וְהִקְרַבְתֶּם - "Howbeit on the tenth day of this seventh month is the day of atonement; there shall be a holy convocation unto you, and ye shall afflict your souls; and ye shall bring an offering made by fire unto the LORD" (Lev. 23:27).
For Christians, who because of Jesus Christ share in the covenant, Hebrews does a nice job of laying out the significance of Yom Kippur:
"Now even the first covenant had regulations for worship and an earthly place of holiness. For a tent was prepared, the first section, in which were the lampstand and the table and the bread of the Presence. It is called the Holy Place. Behind the second curtain was a second section called the Most Holy Place, having the golden altar of incense and the ark of the covenant covered on all sides with gold, in which was a golden urn holding the manna, and Aaron's staff that budded, and the tablets of the covenant. Above it were the cherubim of glory overshadowing the mercy seat. Of these things we cannot now speak in detail.
"These preparations having thus been made, the priests go regularly into the first section, performing their ritual duties, but into the second only the high priest goes, and he but once a year, and not without taking blood, which he offers for himself and for the unintentional sins of the people. By this the Holy Spirit indicates that the way into the holy places is not yet opened as long as the first section is still standing (which is symbolic for the present age). According to this arrangement, gifts and sacrifices are offered that cannot perfect the conscience of the worshiper, but deal only with food and drink and various washings, regulations for the body imposed until the time of reformation.
"But when Christ appeared as a high priest of the good things that have come, then through the greater and more perfect tent (not made with hands, that is, not of this creation) he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption. For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God" (9:1-14- ESV underlining emphasis mine).