What are the "two kinds"?

I think that you are referring to the two kinds of daughters that exist according to Jing-Mei's mother in the story "Two Kinds" by Amy Tan.  

The story is a wonderful read about the dynamic relationship between a mother and a daughter.  Specifically, the story is about the dynamic relationship between Jing-Mei and her mother over the course of a few months.  

Jing-Mei's mother wants Jing-Mei to be a brilliant, famous child prodigy....

I think that you are referring to the two kinds of daughters that exist according to Jing-Mei's mother in the story "Two Kinds" by Amy Tan.  

The story is a wonderful read about the dynamic relationship between a mother and a daughter.  Specifically, the story is about the dynamic relationship between Jing-Mei and her mother over the course of a few months.  

Jing-Mei's mother wants Jing-Mei to be a brilliant, famous child prodigy.  Somebody like Shirley Temple.  

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

For the next few months, the mother pushed Jing-Mei to excel in a number of tests and memorization activities.  Jing-Mei was forced to dress certain ways and do her hair in a specific style.  None of those attempts clearly indicated that Jing-Mei was a future prodigy.  The last big push was to make Jing-Mei into a piano virtuoso.  As time passed though, it became clear to Jing-Mei that she was constantly disappointing her mother; she no longer wanted any part of this prodigy business.  Jing-Mei stopped trying completely, because she realized that she wanted to be herself and not like somebody that her mother wanted her to be like.  

I won't let her change me, I promised myself. I won't be what I'm not.

Jing-Mei's mother did not react well to her daughter's new found courage and obstinate attitude.  She told Jing-Mei that there are only two kinds of daughters in the world.  

"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!"  

Jing-Mei's mother is pushing her daughter to be a more obedient daughter and try harder.  Unfortunately, the two kinds comment has the opposite effect, because Jing-Mei responds in an incredibly hurtful manner.  

"Then I wish I weren't your daughter, I wish you weren't my mother," I shouted. As I said these things I got scared. It felt like worms and toads and slimy things crawling out of my chest, but it also felt good, that this awful side of me had surfaced, at last.

It's the climax of the story.  It's at this point that their relationship is finally strained to the breaking point.  It isn't until years later that each woman is willing to forgive the other.