The term Yom Kippur is actually written in the plural in the Torah, Yom Ha-Kippurim (יום הכפרים), perhaps because the purification process cleansed from a multitude of transgressions, iniquities, and sins.
However, the name also alludes to the two great atonements given by יהוה - the first for those among the nations who turn to Yahusha for cleansing and forgiveness, and the second for the purification of ethnic Yashar'el during Yom Adonai, the great Day of יהוה at the end of days.
Kippurim can be read as Yom Ke-Purim, a "day like Purim," that is, a day of deliverance and salvation (as explained in the Book of Esther). Thus the day on which Yahusha sacrificed Himself on the stake is the greatest "Purim" of all, since through Him we are eternally delivered from the hands of our enemies.
The Torah states that Yom Kippur was the only time when the High Priest could enter the Qadosh haQodeshim to offer blood sacrifice for the sins of the people.
This "life for a life" principle is the foundation of the sacrificial system and marked the great day of intercession made by the High Priest on behalf of Yashar'el.