The "two kinds" that Suyuan is talking about are the two different kinds of daughters that exist.
The line comes fairly late in the story. Suyuan has been pushing Jing-mei for months and months to be some kind of child prodigy. The most recent attempt was to turn Jing-mei into an amazing pianist. It failed miserably because Jing-mei gave it zero effort. In fact, Jing-mei intentionally tried to sabotage the entire endeavor.
So maybe I never really gave myself a fair chance. I did pick up the basics pretty quickly, and I might have become a good pianist at the young age. But I was so determined not to try, not to be anybody different, and I learned to play only the most ear-splitting preludes, the most discordant hymns.
Eventually Jing-mei had to perform in front of a large audience, and it went terribly; however, Jing-mei believed that she had performed poorly enough to cause her mom to stop pushing the piano prodigy dream. It wasn't to be. Two days after the piano recital, Suyuan told Jing-mei to begin practicing. A huge yelling match between mother and daughter ensued. Jing-mei explicitly told her mother that she would not be pushed into becoming something that she wasn't.
"You want me to be something that I'm not!" I sobbed. "I'll never be the kind of daughter you want me to be!"
Suyuan immediately responded with the "two kinds" line.
"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!"
In Suyuan's mind, daughters are either obedient or rebellious. There is no middle ground. The result of the comment is that Jing-mei shouts out that she doesn't want to be Suyuan's daughter anymore. Jing-mei wishes that she were dead like all of her mother's other children. With those things said, the already strained relationship is broken for years to come.