What exactly is Yom Kippur and how is it celebrated?

A well-informed answer from Angel! I’ll just add a couple things: There’s a subtle but very important difference for us between repentance and atonement. Repentance is what we do during the whole preceding month of Elul, leading up to Rosh HaShanah, not so much by breast-beating in self-accusation (one hopes) but by turning our hearts back to G!d. (The Hebrew word t’shuvah that is usually translated as repentance really means “turning.”) Atonement is the state of being in which our relationship with G!d, our spiritual integrity, has actually become restored - you can think of it as “at-one-ment.”

Why does this distinction matter? Because on Yom Kippur we are NOT repenting. Rather, we are enjoying the fruits of our labor of repentance that we have been doing throughout Elul. Our fast on Yom Kippur is not like our other fasts, which ARE fasts of repentance (like on Tisha B’Av, the 9th day of the month of Av, when we mourn the destruction of the ancient Holy Temple in Jerusalem).

On Yom Kippur, in contrast, we fast because our t’shuvah has brought us so close to G!d that we have become like angels who don’t need to eat, but only exist to love and praise G!d like we do all day on Yom Kippur. By nightfall at the end of the day, our tradition teaches us, the day itself (building on the foundation of our own t’shuvah) will have achieved atonement for us.