Leviticus 23:26 etc and is probably the most important holiday of the Jewish year.
Yom Kippur is a day set aside to atone for the sins of the past year. On this day people atone only for sins between man and God, not for sins against another person.
To atone for sins against another person, people must first seek reconciliation with that person, righting the wrongs they committed against the person if possible, and do this before Yom Kippur. This day is the last chance to change the judgment, to demonstrate repentance and make amends.
People are supposed to refrain from eating and drinking (even water) on Yom Kippur. It is a complete, 25-hour fast beginning before sunset on the evening before Yom Kippur and ending after nightfall on the day of Yom Kippur.
The Talmud (the Jewish holy book) also specifies additional restrictions on washing and bathing, anointing one's body (with cosmetics, deodorants, etc.), wearing leather shoes and engaging in sexual relations.
It is customary to wear white on this holiday. White symbolizes purity, and calls to mind the promise that our sins shall be made as white as snow.
Most of the holiday is spent in the synagogue in prayer. The services end at nightfall, with the blowing of a long blast on the shofar or ram's horn.