As a Chinese-American girl whose mother is determined to fulfill her own hopes and dreams by making her daughter a prodigy, Jing-mei struggles to find her own identity. When she rebells against her mother's attempts to make her into something she is not, her mother shouts at her, "Only two kinds of daughters...those who are obedient and those who follow their own mind!"
Jing-mei's mother has decided that her daughter will be an amazing pianist, even though Jing-mei has neither talent nor interest in playing the instrument. She makes the young girl take lessons and practice faithfully, but when Jing-mei's lack of a natural gift in the area is made painfully evident at a performance, Jing-mei refuses to continue playing. Her rebellion has greater significance than a simple refusal to pursue a hobby chosen by her mother, however. Jing-mei's dilemma is central to the theme of the story. Jing-mei must choose whether to obey her mother and continue with the piano, or to listen to her own heart. The choice she makes, as set forth by her mother, will be either good or bad, and reflect Chinese filial expectations vs. American freedom. In making her choice, Jing-mei will choose to be true to her Chinese roots or her American upbringing - she will decide which of "two kinds" of daughters she will be.
The title of this story is very appropriate because it highlights the contrast between Jing-Mei and her mother. On the one hand, Jing-Mei's mother harbors great ambitions for her daughter. She subscribes completely to the idea of the American Dream, that anyone can be anything they want to be in America. This leads her to become obsessed with the idea of turning her daughter into a child prodigy. Some examples include a Chinese version of Shirley Temple and a master on the piano.
In contrast, Jing-Mei is not so absorbed by the idea of the American Dream. In fact, she has no real interest in becoming a child prodigy at all. She is bored by the countless hours of practice and genuinely terrified when her mother tells her that she will start learning the piano:
When my mother told me this, I felt as though I had been sent to hell.
So there are, quite literally, two kinds of beliefs going on in this story. Jing-Mei's mother wants her daughter to be a successful child prodigy, while Jing-Mei simply wants her mother to accept her for who she is. These competing beliefs and desires are what drive the story and create its central conflict.