Yom Kippur is probably the most important holiday of the Jewish year. Many Jews who do not observe any other Jewish custom will refrain from work, fast and/or attend synagogue services on this day. Yom Kippur occurs on the 10th day of Tishri. The holiday is instituted at Leviticus 23:36 et seq.
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.