The fourth out of five major holidays is the Yom ha-Kippurim: lit. the Day of Shelters. Traditionally this holiday is known as the Day of Atonement or Yom Kippur (singular). The word "kippur" literally refers to the Lid of the Ark of the Covenant (i.e Ex. 25:17) and thus the name of the Holiday.
Let's take a look at the main description that can be found in Levtiticus 23:26-32:
26 And the LORD spake unto Moses, saying,
27 Also on the tenth day of this seventh month there shall be a day of atonement: it shall be an holy convocation unto you; and ye shall afflict your souls, and offer an offering made by fire unto the LORD.
28 And ye shall do no work in that same day: for it is a day of atonement, to make an atonement for you before the LORD your God.
29 For whatsoever soul it be that shall not be afflicted in that same day, he shall be cut off from among his people.
30 And whatsoever soul it be that doeth any work in that same day, the same soul will I destroy from among his people.
31 Ye shall do no manner of work: it shall be a statute for ever throughout your generations in all your dwellings.
32 It shall be unto you a sabbath of rest, and ye shall afflict your souls: in the ninth day of the month at even, from even unto even, shall ye celebrate your sabbath.
26 וַיְדַבֵּ֥ר יְהוָ֖ה אֶל־מֹשֶׁ֥ה לֵּאמֹֽר׃27 אַ֡ךְ בֶּעָשׂ֣וֹר לַחֹדֶשׁ֩ הַשְּׁבִיעִ֨י הַזֶּ֜ה י֧וֹם הַכִּפֻּרִ֣ים ה֗וּא מִֽקְרָא־קֹ֙דֶשׁ֙ יִהְיֶ֣ה לָכֶ֔ם וְעִנִּיתֶ֖ם אֶת־נַפְשֹׁתֵיכֶ֑ם וְהִקְרַבְתֶּ֥ם אִשֶּׁ֖ה לַיהוָֽה׃28 וְכָל־מְלָאכָה֙ לֹ֣א תַעֲשׂ֔וּ בְּעֶ֖צֶם הַיּ֣וֹם הַזֶּ֑ה כִּ֣י י֤וֹם כִּפֻּרִים֙ ה֔וּא לְכַפֵּ֣ר עֲלֵיכֶ֔ם לִפְנֵ֖י יְהוָ֥ה אֱלֹהֵיכֶֽם׃29 כִּ֤י כָל־הַנֶּ֙פֶשׁ֙ אֲשֶׁ֣ר לֹֽא־תְעֻנֶּ֔ה בְּעֶ֖צֶם הַיּ֣וֹם הַזֶּ֑ה וְנִכְרְתָ֖ה מֵֽעַמֶּֽיהָ׃30 וְכָל־הַנֶּ֗פֶשׁ אֲשֶׁ֤ר תַּעֲשֶׂה֙ כָּל־מְלָאכָ֔ה בְּעֶ֖צֶם הַיּ֣וֹם הַזֶּ֑ה וְהַֽאֲבַדְתִּ֛י אֶת־הַנֶּ֥פֶשׁ הַהִ֖וא מִקֶּ֥רֶב עַמָּֽהּ׃31 כָּל־מְלָאכָ֖ה לֹ֣א תַעֲשׂ֑וּ חֻקַּ֤ת עוֹלָם֙ לְדֹרֹ֣תֵיכֶ֔ם בְּכֹ֖ל מֹֽשְׁבֹֽתֵיכֶֽם׃32 שַׁבַּ֨ת שַׁבָּת֥וֹן הוּא֙ לָכֶ֔ם וְעִנִּיתֶ֖ם אֶת־נַפְשֹׁתֵיכֶ֑ם בְּתִשְׁעָ֤ה לַחֹ֙דֶשׁ֙ בָּעֶ֔רֶב מֵעֶ֣רֶב עַד־עֶ֔רֶב תִּשְׁבְּת֖וּ שַׁבַּתְּכֶֽם׃ פ
More detailed description can be found in Leviticus 16:1-34, and since its relevant to the Tabernacle, I would like you to follow this brief summary of the holiday proceedings:
The ritual would be performed by the High Priest himself. No one else would be allowed to do it. The priest would have to follow a set of very specific and strict rules so as to avoid any problems such as in the the death of two sons of Aaron (which are mentioned in the first verses of the description on purpose and as a reminder for non-compliance).
I did not research the procedure in detail and maybe mistaken, but it appears that the ritual would begin at sundown (Lev. 23:32):
After initial preparations (Lev. 16:1-11), the High Priest would first fill the Tent of the Tabernacle with Incense fumes (Lev. 16:12-13):
Only then, when Incense smoke would fill and cover (kippur) the entire Tent, only then he would enter the Holy of Hollies section so do the ritualistic blood sprinkling upon the actual "kippur" - the Ark of the Covenants' Lid (Mercy Seat) [Lev. 16:14-15] Thus the name of the Holiday.
He then would sprinkle blood about the rest of the Tent (the Holy Place). He would be alone inside while doing this. (Lev. 16:16-17)
The High Priest would then exit the Tent of the Tabernacle and proceed to atone the Altar of Burnt Offering, sprinkling blood upon its four Horns. (Lev. 16:18-19).
Once he would be done with that, the High Priest would "curse" the so-called chosen "Goat of Departure", the Scapegoat, and assign one of the priests(?) to take it out to the wilderness and release it there. (Lev 16:20-22)
This would signify that the sins are now far away from the congregation. Lev(16:22) Like so:
Then the Priest would conclude the ritual by washing himself in water; the other person who took the Scapegoat out in the wilderness would also be required to wash himself in water. Other miscellaneous procedures would be performed as well and the "humbling of the soul" (fasting) would commence.(Lev. 16:23-34.
Last, but not least, the blow of the Silver Thrumphets would be used to announce the beginning of the holiday:
I hope that this gave you an idea of what the ritual of the Day of Atonement looked at the times of the Tabernacle. But I hope that once I will finish updating my models, I would be able to provide you with more comprehensive review of this interesting ritual.
Have a great Yom ha-Kippurim!