Why does Aunt Alexandra want Atticus to let Calpurnia go?
The answer to this can be found in Chapter 14. The basic reason is that Aunt Alexandra has come to live with the Finches now and she thinks that Scout needs her influence rather than that of Calpurnia.
Aunt Alexandra points out to Atticus that Scout is growing up. She says now that she herself is here to help Scout grow up, they don't need Calpurnia anymore. That is all she actually says, but things that Atticus says a little bit later on make it clear that Alexandra thinks Calpurnia's influence on the kids has not been good enough. She thinks Scout will need her influence from here on out.
I think that the implication is that Alexandra would be better for Scout because they are both white.
When Atticus' sister, Alexandra, decides to come and stay with the family for the duration of the Tom Robinson trial, she decides that Calpurnia is no longer necessary. Alexandra plans to become the female presence in the home; she is a good cook, and she believes she knows what's best for the children (especially Scout, who she feels needs to learn about becoming a "lady"), so she feels Calpurnia is no longer necessary. However, Atticus sets his sister straight.
"Calpurnia's not leaving this house until she wants to... She's a faithful member of this family, and you'll simply have to accept things the way they are."
Her stated reasons are because she is here now and can take care of the kids and every female role that needs to be taken care of. I think a possible underlying and not stated reason is because of Calpurnia's race. She would like to remove Calpurnia from the home so Cal doesn't have to hear some of the discussion that will occur regarding the trial.
Atticus refuses to let her go and lets Alexandra know that Calpurnia is as much of a member of the family as the rest of them. He notes that she has meant so much to the kids and him since their mother passed.