Fasting on Yom Kippur

וְהָיְתָ֥ה לָכֶ֖ם לְחֻקַּ֣ת עוֹלָ֑ם בַּחֹ֣דֶשׁ הַ֠שְּׁבִיעִי בֶּֽעָשׂ֨וֹר לַחֹ֜דֶשׁ תְּעַנּ֣וּ אֶת־נַפְשֹֽׁתֵיכֶ֗ם וְכָל־מְלָאכָה֙ לֹ֣א תַעֲשׂ֔וּ הָֽאֶזְרָ֔ח וְהַגֵּ֖ר הַגָּ֥ר בְּתוֹכְכֶֽם׃

Leviticus 16:29

And this shall be to you a law for all time: In the seventh month, on the tenth day of the month, you shall practice self-denial; and you shall do no manner of work, neither the citizen nor the alien who resides among you.

ויקרא ט״ז:ל׳

כִּי בַיּוֹם הַזֶּה יְכַפֵּר עֲלֵיכֶם לְטַהֵר אֶתְכֶם מִכֹּל חַטֹּאתֵיכֶם לִפְנֵי יְהוָה תִּטְהָרוּ.

Leviticus 16:30

For on this day shall atonement be made for you, to cleanse you; from all your sins shall ye be clean before the LORD.

ויקרא ט״ז:ל״א

שַׁבַּת שַׁבָּתוֹן הִיא לָכֶם וְעִנִּיתֶם אֶת נַפְשֹׁתֵיכֶם חֻקַּת עוֹלָם.

Leviticus 16:31

It is a sabbath of solemn rest unto you, and ye shall afflict your souls; it is a statute for ever.

אבן עזרא על ויקרא ט״ז:כ״ט:ב׳

בחדש השביעי בעשור לחדש תענו את נפשותיכם. אחר שהכתוב אומר ותתענג בדשן נפשכם ידענו כי הענוי הפך התענוג והוא הצום ועוד ונפש נענה תשביע שטעמו כמו ותפק לרעב נפשך כי דרך דברי הנביאים להכפל ואחר שיש לנו קבלה אין צורך לחפש ואין עניתי בצום נפשי ראיה בעבור זכר צום והכלל כל עינוי שימצא במקרא דבק עם נפש הוא הצום:

Ibn Ezra on Leviticus 16:29:2

in the seventh month, on the tenth of the month, you will afflict your souls From the passage “let your soul delight in fatness” [Isaiah 55:2] we know that “affliction” of a soul must be the opposite of its delight — i.e., a fast. Moreover, the phrase “be generous to the hungry” [Isaiah 58:10] means the same as “satisfy the afflicted soul” [Isaiah 58:10], because the prophetic style is to speak in parallelisms. Since we possess a tradition for the meaning of this idiom, there is no need to adduce additional evidence (the phrase “I afflicted my soul with fasting” [Psalms 35:13] does not provide additional evidence, because there the word “fasting” is explicitly mentioned). The general rule is: “affliction of soul” denotes fasting, wherever it appears in Scripture.

משנה יומא ח׳:א׳

(א) יום הכפורים אסור באכילה. ובשתיה. וברחיצה. ובסיכה. ובנעילת הסנדל. ובתשמיש המטה.

Mishnah Yoma 8:1

(1) On Yom Kippur, it is forbidden to eat, to drink, to wash, to anoint, and to wear leather shoes and to have sexual relations.

ישעיהו נ״ח:ג׳-י׳

(ג) לָ֤מָּה צַּ֙מְנוּ֙ וְלֹ֣א רָאִ֔יתָ עִנִּ֥ינוּ נַפְשֵׁ֖נוּ וְלֹ֣א תֵדָ֑ע הֵ֣ן בְּי֤וֹם צֹֽמְכֶם֙ תִּמְצְאוּ־חֵ֔פֶץ וְכָל־עַצְּבֵיכֶ֖ם תִּנְגֹּֽשׂוּ׃

(ד) הֵ֣ן לְרִ֤יב וּמַצָּה֙ תָּצ֔וּמוּ וּלְהַכּ֖וֹת בְּאֶגְרֹ֣ף רֶ֑שַׁע לֹא־תָצ֣וּמוּ כַיּ֔וֹם לְהַשְׁמִ֥יעַ בַּמָּר֖וֹם קוֹלְכֶֽם׃

(ה) הֲכָזֶ֗ה יִֽהְיֶה֙ צ֣וֹם אֶבְחָרֵ֔הוּ י֛וֹם עַנּ֥וֹת אָדָ֖ם נַפְשׁ֑וֹ הֲלָכֹ֨ף כְּאַגְמֹ֜ן רֹאשׁ֗וֹ וְשַׂ֤ק וָאֵ֙פֶר֙ יַצִּ֔יעַ הֲלָזֶה֙ תִּקְרָא־צ֔וֹם וְי֥וֹם רָצ֖וֹן לַיהוָֽה׃

(ו) הֲל֣וֹא זֶה֮ צ֣וֹם אֶבְחָרֵהוּ֒ פַּתֵּ֙חַ֙ חַרְצֻבּ֣וֹת רֶ֔שַׁע הַתֵּ֖ר אֲגֻדּ֣וֹת מוֹטָ֑ה וְשַׁלַּ֤ח רְצוּצִים֙ חָפְשִׁ֔ים וְכָל־מוֹטָ֖ה תְּנַתֵּֽקוּ׃

(ז) הֲל֨וֹא פָרֹ֤ס לָֽרָעֵב֙ לַחְמֶ֔ךָ וַעֲנִיִּ֥ים מְרוּדִ֖ים תָּ֣בִיא בָ֑יִת כִּֽי־תִרְאֶ֤ה עָרֹם֙ וְכִסִּית֔וֹ וּמִבְּשָׂרְךָ֖ לֹ֥א תִתְעַלָּֽם׃

(ח) אָ֣ז יִבָּקַ֤ע כַּשַּׁ֙חַר֙ אוֹרֶ֔ךָ וַאֲרֻכָתְךָ֖ מְהֵרָ֣ה תִצְמָ֑ח וְהָלַ֤ךְ לְפָנֶ֙יךָ֙ צִדְקֶ֔ךָ כְּב֥וֹד יְהוָ֖ה יַאַסְפֶֽךָ׃

(ט) אָ֤ז תִּקְרָא֙ וַיהוָ֣ה יַעֲנֶ֔ה תְּשַׁוַּ֖ע וְיֹאמַ֣ר הִנֵּ֑נִי אִם־תָּסִ֤יר מִתּֽוֹכְךָ֙ מוֹטָ֔ה שְׁלַ֥ח אֶצְבַּ֖ע וְדַבֶּר־אָֽוֶן׃ (י) וְתָפֵ֤ק לָֽרָעֵב֙ נַפְשֶׁ֔ךָ וְנֶ֥פֶשׁ נַעֲנָ֖ה תַּשְׂבִּ֑יעַ וְזָרַ֤ח בַּחֹ֙שֶׁךְ֙ אוֹרֶ֔ךָ וַאֲפֵלָתְךָ֖ כַּֽצָּהֳרָֽיִם׃

Isaiah 58:3-10

(3) “Why, when we fasted, did You not see? When we starved our bodies, did You pay no heed?”

Because on your fast day you see to your business and oppress all your labourers! (4) Because you fast in strife and contention, and you strike with a wicked fist! Your fasting today is not such as to make your voice heard on high.

(5) Is such the fast I desire, a day for men to starve their bodies? Is it bowing the head like a bulrush and lying in sackcloth and ashes? Do you call that a fast, a day when the LORD is favorable?

(6) No, this is the fast I desire: to unlock fetters of wickedness, and untie the cords of the yoke. To let the oppressed go free; to break off every yoke.

(7) It is to share your bread with the hungry, and to take the wretched poor into your home; when you see the naked, to clothe him, and not to ignore your own kin.

(8) Then shall your light burst through like the dawn and your healing spring up quickly; your Vindicator shall march before you, the Presence of the LORD shall be your rear guard.

(9) Then, when you call, the LORD will answer; when you cry, He will say: Here I am. If you banish the yoke from your midst, the menacing hand, and evil speech, (10) And you offer your compassion to the hungry and satisfy the famished creature - then shall your light shine in darkness, and your gloom shall be like noonday.

Adin Steinsaltz, Teshuvah: A Guide for the Newly Observant Jew, pp 3-4.
Broadly defined, teshuvah is more than just repentance from sin; it is a spiritual reawakening, a desire to strengthen the connection between oneself and the sacred... All forms of teshuvah, however diverse and complex, have a common core: the belief that human beings have it in their power to effect inward change.

What is a "sin?"
The Hebrew word cheit, often translated as "sin," is from a root meaning "to miss the mark," as in archery or stone-throwing. Many commentators have drawn important implications from the etymology of cheit: As with a stone thrower or archer, our intent is to aim true and to do the right thing; wrongdoing does not cause an ineradicable strain. With practice and attention, we can improve our aim and do better in the future.