The swan feather

From the book:
But when she arrived at the new country, the immigration officials pulled her swan away from her, leaving the old woman fluttering her arms and with only one swan feather for a memory. And then she had to fill out so many forms she forgot why she had come and what she had left behind.

Now the woman was old. And she had a daughter who grew up speaking only English and swallowing more Coca-Cola than sorrow. For a long time now the woman had wanted to give her daughter the single feather and tell her, "This feather may look worthless, but it comes from afar and carries with it all my good intentions." And she waited, year after year, for the day she could tell her daughter this in perfect American English.

As Ali says the swan feather represents the good intentions the four mothers are trying to give their daughters. Such as pushing June to play the piano and Waverly to play chess. They want their daughters to know why they had done the things they had done.
"The woman had to fill out so many forms she had forgotten why she had come." the mothers do not want their daughters to forget their heritage, where the mothers had come from and the hardships they had to endure to get where they are today. But the daughters had swallowed too much Coca Cola and become so Americanized that they really didn't hear the message until this meeting of the Joy Luck Club. Now each of the daughters are understanding their own lives and sorrows are actually mimicking those each of their mothers endured in China. They are now able to take a stand and do something about it... Rose confronted her husband about the affair and Lena discovered her self worth and left her husband to start again.