In Biblical Hebrew, the Day of Atonement is called “Yom Ha-kippurim”; colloquially, we call it “Yom Kippur.” The root of “kippurim,” K-P-R, has many meanings, all of which are relevant to Yom Kippur. First, “kapara” can indicate covering. This was the function of the kaporet, the lid that covered the ark in the Temple; likewise, kapara covers up sins. A second meaning of “kapara” is an exchange or ransom (kofer). Sin, which originates in impure forces, is “exchanged” and returned to its place via the goat of Azazel. Third, “kapara” can mean cleansing or wiping clean. Atonement wipes us clean of the foulness of sin. Fourth, “kapara” can indicate neutralization or renouncement. As Yaakov says, “I will neutralize (akhapra) [Esav’s anger]” (Bereishit 32:21); Yaakov sought to counter Esav’s wrath with gifts (Rashi ad loc.). Fifth, “kapara” indicates appeasement. Neutralizing a sin can appease an injured party or heavenly accuser. (See Rashi, Mishlei 16:14.) Finally, “kapara” can connote fragrance, as one might understand the phrase “eshkol ha-kofer” (Shir Ha-shirim 1:14) to mean “a spray of fragrant blooms.” So too, repentance out of love transforms unknowing sins into merits, releasing a pleasant scent.
Commenting on the verse, “Days will be created, and one of them will be His” (Tehilim 139:16), our Sages suggest that “one of them” refers to Yom Kippur, the most unique day of the year:
For it is a happy time for God, Whose word brought the world into being, Who lovingly gave [Yom Kippur] to the Jewish people. This can be compared to a king of flesh and blood whose servants and household members collected the palace garbage and threw it in front of his doorway (in order to remove it from the city). When the king saw the garbage, it made him very happy. Similarly, God gave Yom Kippur to the Jews out of love and with great joy…. When He forgives the sins of Israel, He is not sad, but rather very happy. He tells the mountains and hills, the rivers and valleys: “Celebrate with Me! Let all rejoice, for I am forgiving Israel’s sins!” (Tanna De-vei Eliyahu Rabba 1)
Our Sages tell us that Yom Kippur’s special power is hinted at in the name of the accusing angel – ha-satan – whose name has the numerical value of 364. For 364 days of the year, the prosecuting angel is permitted to block the divine light from manifesting in the world and to prosecute the Jewish people. But a year has 365 days, so there is one day a year on which the satan may not prosecute Israel: Yom Kippur, when root of the sanctity of Israel, connected with God, is revealed (Yoma 20a). Had God not established a day of atonement and forgiveness, sins would accumulate year after year. Eventually, Israel and the whole world would deserve destruction. (See Sefer Ha-ḥinukh §185.)
The entire day of Yom Kippur atones. Therefore, even if someone dies during Yom Kippur, his sins are forgiven (Keritot 7a). However, atonement is completed at the end of Yom Kippur, when the fast reaches its peak and all the prayers and meditations of Yom Kippur merge together (y. Yoma 4:7). This accounts for the intensity of Ne’ila, the closing prayer.