In this article, I'd like to look into the first Mishna's of Masechet Yoma, which could reveal much of the essence and meaning of Yom Kippur. We will see an interesting comparison between Yom Kippur and the incident of Nadav and Avihu.
The first Mishna of Masechet Yoma opens up with the description what the Kohen Gadol needs to do a week before Yom Kippur:
שבעת ימים קודם יום הכיפורים מפרישין כהן גדול מביתו ללשכת פלהדרין
"Seven days prior to Yom Kippur the Sages would remove the High Priest, who performs the entire Yom Kippur service, from his house to the Chamber of Parhedrin, a room in the Temple designated specifically for the High Priest during that period."
What is the source of this commandment?
The G'mara (Yoma, 2, b) brings that this is taught from the Pasuk (Vayikra, 8, 34) describing the procedure that was necessary before the "Chanukat Hamishkan":
" וּמִפֶּתַח אֹהֶל מוֹעֵד לֹא תֵצְאוּ שִׁבְעַת יָמִים עַד יוֹם מְלֹאת יְמֵי מִלֻּאֵיכֶם כִּי שִׁבְעַת יָמִים יְמַלֵּא אֶת יֶדְכֶם:(לד) כַּאֲשֶׁר עָשָׂה בַּיּוֹם הַזֶּה צִוָּה ה' לַעֲשֹׂת לְכַפֵּר עֲלֵיכֶם:(לה) וּפֶתַח אֹהֶל מוֹעֵד תֵּשְׁבוּ יוֹמָם וָלַיְלָה שִׁבְעַת יָמִים וּשְׁמַרְתֶּם אֶת מִשְׁמֶרֶת ה' וְלֹא תָמוּתוּ כִּי כֵן צֻוֵּיתִי"
"You shall not go outside the entrance of the Tent of Meeting for seven days, until the day that your period of ordination is completed. Your ordination will require seven days.
Everything done today, Hashem has commanded to be done [seven days], to make expiation for you."
Since we should assume that the commandments and their sources aren't random, seemingly there is a connection between The day of Chanukat Hamishkan and the holy day of Yom Kippur. On both the Kohen Gadol is required to prepare himself (and so are we) to the holy day. In addition to this, I'd like to suggest that perhaps every Yom Kippur itself is a kind of reconstruction and rehearsal of the Yom Chanukat Hamishkan.
This relates to the clear connection between Yom Kippur and "Yom Hashmini" in the Torah (the eight day):
" וַיְהִי בַּיּוֹם הַשְּׁמִינִי קָרָא מֹשֶׁה לְאַהֲרֹן וּלְבָנָיו וּלְזִקְנֵי יִשְׂרָאֵל"
"On the eighth day Moses called Aaron and his sons, and the elders of Israel."
And later on, after the incident of the Eight day with Nadav and Avihu, comes the commandment (Vayikra 16, 1-2) for Yom Kippur services:
" וַיְדַבֵּר ה' אֶל מֹשֶׁה אַחֲרֵי מוֹת שְׁנֵי בְּנֵי אַהֲרֹן בְּקָרְבָתָם לִפְנֵי ה' וַיָּמֻתוּ: (ב) וַיֹּאמֶר ה' אֶל מֹשֶׁה דַּבֵּר אֶל אַהֲרֹן אָחִיךָ וְאַל יָבֹא בְכָל עֵת אֶל הַקֹּדֶשׁ מִבֵּית לַפָּרֹכֶת אֶל פְּנֵי הַכַּפֹּרֶת אֲשֶׁר עַל הָאָרֹן וְלֹא יָמוּת כִּי בֶּעָנָן אֵרָאֶה עַל הַכַּפֹּרֶת:"
Hashem spoke to Moses after the death of the two sons of Aaron who died when they drew too close to the presence of Hashem,
Hashem said to Moses: Tell your brother Aaron that he is not to come at will into the Shrine behind the curtain, in front of the cover that is upon the ark, lest he die; for I appear in the cloud over the cover."
The Ramban brings (Vayikra 16) that the commandment for Yom Kippur came immediately after the incident on Yom Hashmini where the two sons of Ahron died offering a "Eish Zara".
Their sin described in Parshat shemini (Vayikra 10,1):
" וַיִּקְחוּ בְנֵי אַהֲרֹן נָדָב וַאֲבִיהוּא אִישׁ מַחְתָּתוֹ וַיִּתְּנוּ בָהֵן אֵשׁ וַיָּשִׂימוּ עָלֶיהָ קְטֹרֶת וַיַּקְרִיבוּ לִפְנֵי ה' אֵשׁ זָרָה אֲשֶׁר לֹא צִוָּה אֹתָם"
"Now Aaron’s sons Nadab and Abihu each took his fire pan, put fire in it, and laid incense on it; and they offered before Hashem alien fire, which had not been enjoined upon them."
In this matter, Yom Kippur is trying to accomplish a "take 2" for the Yom Hashmini and perhaps an improvement and correction of the death of Nadav and Avihu.
This comparison is strengthened by a few parables we find between the obligations of Yom Kippur and the death of Nadav and Avihu.
In this same first Mishna of the Masechet, it is recited that the Kohen Gadol is put in a room called "Lishkat Parhedrin". The G'mara explains (Yoma, 8, b), that this is named after the fact that many of them would die every year and be switched. This reminds us of Nadav and Avihu who died while entering and offering an improper offer of incense.
The next Sentence of our Mishna also relates to this:
"רבי יהודה אומר אף אשה אחרת מתקינין לו שמא תמות אשתו שנאמר וכפר בעדו ובעד ביתו ביתו זו אשתו אמרו לו
אם כן אין לדבר סוף"
"Rabbi Yehuda says: The Sages would even designate another wife for him lest his wife die, as it is stated in the Torah portion of the Yom Kippur service: “And it will atone for him and for his house” (Leviticus 16:6); the Sages interpreted the term: His house, that is his wife. The priest must be married in order to fulfill this commandment. Due to the concern lest his wife die, another wife was designated to address that possibility. The Rabbis said to Rabbi Yehuda: If so, that this is a concern, there is no end to the matter, as what if the designated replacement wife dies? This possibility need not be a source of concern."
According to the Mishna, a Kohen Gadol must be married on Yom Kippur. The Rashar Hirsh (Vayikra, 16, 6) explains that this requirement is showing the remodel of Judaism, how our holiest person is connected to earthly matters and family. Interestingly, one of the reasons explaining Nadav and Avihu sin is the fact that they weren't married (Vayikra Rabbah, Acharei Mot, 20,9).
In the next Mishna (Yoma, 18, b), we read about how the Beis Din would need to swear the Kohen Gadol that he doesn't change anything in the process of offering the inner incense while entering the Kodesh Hakodashim. The Gmara explains (Yoma,19, b) that there was a concern that the Kohen Gadol belonged to the cult or group of Tzedokim that would intentionally try and change the order of the incense, by first putting the incense on the coal and only then entering the Kodesh Hakodashim, instead of going according to Chazzal who ruled that the incense is placed on the coal only after already entering the Kodesh hakodashim. The Rema Mipano (Asara Ma'amarot, Chikur Din,15) writes that Nadav and Avihu sin was that they offered the incense in the same way as the Tzedokim here. Here we see another parable between the two cases.
Another parable is the way and character of the day. The Kli Yakar (Acharei Mot) points out the fact that on Yom Kippur we all fast and refrain from all eating and drinking. Unlike so, Nadav and Avihu acted differently. During The giving of the Torah at Mount Sinai, the Torah (Shmot 24, 9-11) descries:
" וַיַּעַל מֹשֶׁה וְאַהֲרֹן נָדָב וַאֲבִיהוּא וְשִׁבְעִים מִזִּקְנֵי יִשְׂרָאֵל:(י) וַיִּרְאוּ אֵת אֱלֹהֵי יִשְׂרָאֵל וְתַחַת רַגְלָיו כְּמַעֲשֵׂה לִבְנַת הַסַּפִּיר"
וּכְעֶצֶם הַשָּׁמַיִם לָטֹהַר:(יא) וְאֶל אֲצִילֵי בְּנֵי יִשְׂרָאֵל לֹא שָׁלַח יָדוֹ וַיֶּחֱזוּ אֶת הָאֱלֹהִים וַיֹּאכְלוּ וַיִּשְׁתּוּ:
"and they saw the God of Israel—under whose feet was the likeness of a pavement of sapphire, like the very sky for purity.
Yet [God] did not raise a hand against the leaders of the Israelites; they beheld God, and they ate and drank."
And Rashi explains:
רש"י:
ויראו את אלהי ישראל - נסתכלו והציצו ונתחייבו מיתה, אלא שלא רצה הקדוש ברוך הוא לערבב שמחת התורה, והמתין לנדב ואביהוא עד יום חנוכת המשכן...(יא) ואל אצילי - הם נדב ואביהוא והזקנים: לא שלח ידו - מכלל שהיו ראוים להשתלח בהם יד:ויחזו את האלהים - היו מסתכלין בו בלב גס מתוך אכילה ושתיה, כך מדרש תנחומא.
"They gazed at him intimately as though their association with Him were a matter of eating and drinking."
In fact, these people were Nadav and Avihu. Rashi brings (Shmot, 24,11) that they were actually supposed to die then, but Hashem waited with it and postponed it till the Yom Hashmini. The Kli Yakar explains that that's the reason why we fast on Yom Kippur, to correct their misconception. We also know that according to one opinion (Vayikra Rabbah, 20, 9), their sin was by entering the Kodesh after they have drunk wine. Nevertheless, The Kohen Gadol was not allowed to eat much before the fast (Yoma 18, b).
Another two opinions (Vayikra Rabbah, 20, 9) explaining their sin, is that they entered without fully wearing the Bigdei Kehuna or not having been washed hands and feet. Interestingly, on Yom Kippur the Kohen Gadol must change his clothes five times (!) and wash his hands and feet ten (!) times. Clearly very different from Nadav and Avihu's behavior
The meaning of Yom Kippur
The Torat Shalem (Shmini, 12) explains, that Nadav and Avihu wanted to serve Hashem extremely without and intermediates:
ספר תורת שלום, פרשת שמיני, אות יב':
דהנה בצפנת פענח על התורה פרשת שמיני כתב דחטאם היה שלא נתנו בה מעלה עשן...והם לשיטתם דס"ל דאין אנו צריכים לאמצעים וכמ"ש בדרוש בגדר ותחת רגליו כמעשה לבנת הספיר...וזה הסמיכות ויראו את האלקים ויאכלו וישתו שבא הכל לשיטתם שאין צריך לאמצעי ואז לא נענשו שלא רצה הקב"ה לערבב השמחה כמ"ש ברש"י שם, לזה אמר כאן שלא כן הוא רק ביוכ"פ לבד אז אין צריך לאמצעי אבל לא בשאר ימות השנה והבן. וגם נ"ל לתת טעם למ"ש בעשרה מאמרות מאמר חקור דין (ח"ד פט"ו) שחטאם היה שתקנו בחוץ והכניסו בפנים כדרך הצדוקים שהם דרשו כי בענן אראה מלמד שיתקן מבחוץ ויכניס, ואנן אמרינן ביומא (נ"ג) שבא לומר שניתן בו מעלה עשן, וכיון שהם לא נתנו מעלה עשן לכן תקנו מבחוץ.
This could explain why they entered the way they did. According to this, Yom Kippur is meant to correct this. On one hand we DO want to enter, and the whole purpose of the day is to stand "Lefnei Hashem"- Before Hashem. On the other hand, we want to do it right. So fixing something doesn't always mean that the first approach was completely wrong. There was truth to it, and Yom Kippur teaches us how to redo the Yom Hashmini correctly. That could explain why the Arizal says (Brought in the Mishnah Berurah, Yom Kippur, Siman 621, b), that anyone who cries over the death of the two sons of ahron on Yom Kippur has all his sins forgiven. Perhaps because he understands how to continue their desire properly.