When Scout and Jem arrive home from Calpurnia's church at the end of Ch. 12, they find Aunt Alexandra sitting in a rocking chair on the porch. They ask her if she's come for a visit, but her reply is much more discouraging (at least for Scout).
"Well, your father and I decided it was time I came to stay with you for a while."
"For a while" in Maycomb meant anything from three days to thirty years. Jem and I exchanged glances.
So obviously, neither Finch child is really looking forward to having their aunt stay for any length of time. Then she reveals her true purpose:
"Jem's growing up now and you are too," she said to me. "We decided that it would be best for you to have some feminine influence. It won't be many years, Jean Louise, before you become interested in clothes and boys--"
So, it seems that Atticus and Aunt Alexandra have decided that Scout needs "feminine influence". That is, she's going to teach Scout how to be a girl, in Aunt Alexandra's definition of the term. Knowing Aunt Alexandra's personality, it's more likely that she decided Scout needed a woman's guidance, & Atticus couldn't say no. The fact that Alexandra doesn't consider Calpurnia a suitable female role model for Scout also reveals much about her character. With Alexandra's emphasis on status and family history, she has no faith in Calpurnia, because she's black and in the position of a servant in the Finch household. Although all the Finches (Atticus included) consider Calpurnia as a second mother to Jem and Scout, Aunt Alexandra simply cannot accept this. So, she makes herself a place in their lives, for better or worse.
In Chapter 12, Jem and Scout go with Calpurnia to her church. When they come back home, they see their Aunt Alexandra waiting for them on the porch. She has come to Maycomb to live with Atticus and the two kids.
In Chapter 13, we find out pretty much why she has come. Basically, what is going on is that she thinks the kids are not really being raised properly. She thinks they need a white woman to raise them (rather than Calpurnia and Atticus). She also thinks that they need to be more aware of their (rather high) place in Maycomb society. They need to start acting more like they are part of the elite.
So Aunt Alexandra is basically there to reform the way the kids are being brought up.