Yom Kippur is the Day of Atonement. In ancient times, it was more festive than now and unmarried people of both genders would meet and try to find their future spouses.
Yom Kippur is a serious but also a happy day, since all Jews who repent (do teshuva) have their sins forgiven on that day.
Teshuva has four main parts:

• Charata  Regretting what we have done and feeling bad about it.
• Vidui  Recognizing and admitting that we have done something wrong.
•  Kabala  Resolving not to repeat that mistake.
• Azivat hachet  Being in the same situation as before but avoiding doing the sin.

The Torah tells us that there is an inherent property to Yom Kippur that causes spiritual purification and removes sins.
All Jews can become like angels on Yom Kippur.  In order to imitate angels (which are spiritual beings), we abstain from five activities that are associated with physical beings. We do not wash, anoint our bodies, eat or drink, have intimate relations, or wear leather shoes. To further imitate angels, when we say the shema, we say Baruch shem kevod malchuto l'olam va'ed out loud, as opposed to the rest of the year, when we say it quietly.
For more on fast days, see Fast Days.