In "Two Kinds", how is Tan's use of authentic Chinese language significant in the story?
Your original question didn't specify which "story" you wished to focus on, so I have tried to guess and gone for "Two Kinds." However, in my response I will keep my comments general. What I say equally applies to all of Tan's fiction, so hopefully you will be able to get something out of it.
Tan's fiction mostly focuses on the generation gap between first-generation Chinese immigrants and second-generation Chinese immigrants. By this I mean immigrants that have left China and gone to the United States (first-generation Chinese immigrants) and then their children, who are Chinese yet not because they are born in the United States (second generation immigrants). Tan writes amazing fiction that concerns itself with the conflict between these two groups of people as the parents have their Chinese values and hopes for their children and their children have their own different values shaped by their identity.
In "Two Kinds", then, the mother's English is not very good, and the use of authentic Chinese is used by Tan to highlight her identity as a first-generation migrant who in a sense is still more Chinese than she is American. The daughter, Jing-Mei, speaks flawless English, and thus the divide is highlighted.
Hope this helps give you some general pointers about how the use of Chinese is utilised by Tan in her fiction.