Yom Kippur, the Sabbath of Sabbaths, is observed the tenth day of the Hebrew month of Tishri, September or early October in the secular calendar.
The first Yom Kippur took place after Moses returned from his second trip to Mt. Sinai with the replacement set of tablets containing the Ten Commandments. He had broken the original set when he returned the first time to discover the children of Israel worshipping a golden calf rather than G-d, who brought them out from Egypt.
Moses successfully pleaded with G-d on their behalf, and on the first of Elul, he ascended the mountain, this time for a second set of tablets. In Moses' absence, the nation fasted from sunrise to sunset. Moses descended the mountain on the tenth of Tishri. Upon returning, Moses found the nation truly repentant and announced that G-d had forgiven them. He decreed that the tenth of Tishri would remain a day of atonement for all generations.
And this shall be an eternal law for you. Each year on the tenth day of the seventh month you must fast and do no work. This is true of the native born and of the proselyte who comes to join you. This is because on this day you shall have all your sins atoned, so that you will be cleansed. Before G-d you will be cleansed of your sins. It is a Sabbath of Sabbaths to you, and a day upon which you must fast. This is a law for all time. (Leviticus 16:29-31)