What does the title "Two Kinds" mean? What are there two kinds of? What do those two kinds have to do with the piano pieces "Pleading Child" and "Perfectly Contented"?

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The title refers to Jing-Mei’s epiphany about mother-daughter relationships.  Jing-mei remembers two different songs, when in fact they are part of the same song.  The song is a metaphor for childhood, because we are sometimes content and sometimes pleading, and childhood is about balance.

Jing-Mei’s mother wants her to become a prodigy.  She is more interested in accomplishment than fame.  She attempts several television-inspired talents.  None of them take.  The result is just Jing-mei and her mother’s frustration.

I looked at my reflection, blinking so that I could see more clearly. The girl staring back at me was angry, powerful. She and I were the same. I had new thoughts, willful thoughts - or rather, thoughts filled with lots of won'ts. I won't let her change me, I promised myself. I won't be what I'm not.

They settle on the piano, but that’s a disaster too.  Jing-Mei’s teacher is deaf and not really paying attention.  She does not really learn how to play the piano.  When the recital comes, she embarrasses herself by not really being able to play.

Jing-Mei and her mother have a rocky relationship, and it comes to a head over the whole prodigy thing.  Jing-Mei’s mother wants her to do what she says, and Jing-mei wants to be her own person.

"Only two kinds of daughters," she shouted in Chinese. "Those who are obedient and those who follow their own mind! Only one kind of daughter can live in this house. Obedient daughter!"

"Then I wish I weren't your daughter, I wish you weren't my mother," I shouted.

It is from this comment that we get the title of the chapter.  Jing-mei's mother wants her to be obedient, but Jing-mei feels like she is the other kind of daughter.  There is no middle ground, and therefore they have a relationship full of conflict.

When Jing-mei’s mother dies, she goes back to the piano and looks at it.  Seeing the music, she notices the name of the song.  As a child, she did not realize that “Pleading Child” and “Contented Child” were two parts of the same song.  It is a metaphor for childhood.  All children need to negotiate the emotional times in their lives, and there will be ups and downs.