This is a Yin Yang sign. This symbol symbolizes the understands of how everything works. This symbol is also known as the Tai-Chi symbol. One of the reasons why the symbol was created was to show how the world needs both positivity and masculine and negativity and feminine to survive. It shows that both are balanced and neither one has more then the other. Yin and Yang are opposites but the circle shows how they get along.
This is important in Joy Luck Club because it shows how everything should be balanced and that you need one to go with the other meaning the American and Chinese different ways. It shows mostly through the younger generation because of the gap.