Jing-Mei says this of her cousin Waverly:

We had grown up together and shared all the closeness of two sisters, squabbling over crayons and dolls. In other words, for the most part, we hated each other. I thought she was snotty. Waverly Jong had gained a certain amount of fame as "Chinatown's Littlest Chinese Chess Champion." "She bring home too many trophy." Auntie Lindo lamented that Sunday. "All day she play chess. All day I have no time do nothing but dust off her winnings."

Waverly represents the Asian dream of success in America, the dream that states you can excel and be whatever you want to be. Jing-Mei knows her mother wants her to have that kind of competitive success at music and so rebels. She wants to be her own person, not what her mother wants to mold her into.

Jing-Mei plays the wrong notes and has a terrible piano recital at a talent show. She then fights bitterly with her mother over continuing practicing the piano. Her mother says she must be an "obedient" child but Jing-Mei goes her own way. Waverly, at least as presented in this story, fulfills the role of the obedient child and feels superior to Jing-Mei for doing so. Waverly represents the obedience that wars internally with Jing-Mei's more free spirited personality.

In the story "Two Kinds" by Amy Tan, what is the conflict between Jing-mei and her mother?

The conflict between Jing-mei and her mother is over Jing-mei’s piano lessons.  Jing-mei does not want to play anymore, but her mother wants her to become a prodigy.

When Jing-mei’s mother decides she should be a prodigy, Jing-mei is on board at first.

"Of course, you can be a prodigy, too," my mother told me when I was nine. "You can be best anything…”

Jing-mei is interested in being a prodigy.  She plays along for a while, trying to become some kind of prodigy.  It doesn’t take, and Jing-mei gets frustrated.  When her mother tries to get her to take piano lessons, she is not excited.

"Why don't you like me the way I am?" I cried. "I'm not a genius! I can't play the piano. And even if I could, I wouldn't go on TV if you paid me a million dollars!"

But Jing-mei’s mother insists, and she starts the lessons with the elderly—and deaf—Mr. Chong.  Unfortunately, she figures out that she does not really have to learn anything pretty quickly.  As a result, her first concert is an embarrassment for Jing-mei and her mother.

Jing-mei and her mother have a big argument when her mother tries to get her to continue playing the piano after the disastrous conflict.  Her mother drags her to the piano bench to practice, and Jing-mei tells her she wishes she was dead like her twin babies in China.  Her mother is so saddened that she drops the issue.