Yom Kippur is the holiest day on the Jewish calendar and begins today at sundown. Also known as "The Day of Atonement," during this holiday Jews focus on teshuvah (repentance), prayer and fasting. The purpose of these actions - all of which are viewed in a joyous light - is to bring about reconciliation between people and between individuals and God.
Wondering how to greet your Jewish friends during Yom Kippur? Since most Jews will fast during this holiday it is appropriate to wish them an "Easy Fast," or "Tzom Kal" in Hebrew. You can also use the traditional Yom Kippur greeting of "G'mar Hatimah Tovah," which means "May you be sealed for a good year (in the Book of Life)." This greeting reflects the view that God seals our fates on Yom Kippur and inscribes our names in either the Book of Life or the Book of Death. Finally, since Rosh HaShanah through Yom Kippur is viewed as the beginning of the New Year, you can also simply say "L'Shanah Tovah!" ("Happy New Year!")
Learn more about Yom Kippur and its meaning in: What Is Yom Kippur?