Yom Kippur, also known as the Day of Atonement, is the holiest day of the year in Judaism. Its central themes are atonement and repentance. Jews traditionally observe this holy day with an approximate 25-hour period of fasting and intensive prayer, often spending most of the day in synagogue services.
The name "Yom Kippur" means "Day of Atonement," and that pretty much explains what the holiday is. It is a day set aside to "afflict the soul," to atone for the sins of the past year. In Days of Awe, I mentioned the "books" in which G-d inscribes all of our names. On Yom Kippur, the judgment entered in these books is sealed. This day is, essentially, your last appeal, your last chance to change the judgment, to demonstrate your repentance and make amends.
As I noted in Days of Awe, Yom Kippur atones only for sins between man and G-d, not for sins against another person. To atone for sins against another person, you must first seek reconciliation with that person, righting the wrongs you committed against them if possible. That must all be done before Yom Kippur.