In mainland China, the festival is celebrated by families gathering together for a meal, giving thanks for the harvest and to ask for blessings (i.e. babies, spouse, beauty, good health, etc).
A key component of this festival is the moon. The Chinese believe the moon and water help with rejuvenation. There is a legend taht the sun and moon are married and the stars are their children. When there is a full moon, it means the moon is pregnant and when it is a crescent shape she has given birth. This belief makes it a popular holiday for women to worship and give offerings to the moon.
It is popular for people to give the gift of moon cake to friends and family. It is a few inches in diameter and has a stamped picture on top. They come in bright, beautiful packages. The moon cake we received had a dragon on it. The closest thing we have to moon cake in the USA are fig newtons. Moon cakes have different fillings inside: red bean (which the Chinese consider very sweet and delicious), green bean, and egg. When I lived in Hong Kong I liked the pineapple flavor but they aren't sold in Xi'an.
Aida, a Chinese friend, told me that to the Chinese circles represent completeness and unity. So sharing of a moon cake with family represents sharing and unity of a family.
My kids have been looking forward to trying moon cake for several weeks, however our tastes are still pretty American and they didn't enjoy the cake.