In considering this question, it is important to understand the difference between "tone" and "mood."
The tone is how the author feels about his/her subject. The mood is how the author wants the reader to feel. Sometimes they are the same in a story, but not always.
In Harper Lee's To Kill a Mockingbird, the tone of Chapter 13 reflects the author's difficulty with drawing one's sense of value from social standing (Aunt Alexandra would have the children act) as opposed to the value of one's character, something Atticus teaches his children and demonstrates by example.
Atticus has raised his children well. They get along with others not based upon what part of town they live in or how much money their parents make, but based upon their value as people. For example, while the Cunninghams don't have money, Scout is not disrespectful of them. She understands from her father's explanations that one needs to find a place to view life through another's eyes to fully appreciate him or her. In Chapter Three, Atticus explains:
"First of all," he said, "if you can learn a simple trick, Scout, you'll get along a lot better with all kinds of folks. You never really understand a person until you consider things form his point of view—"
"—until you climb into his skin and walk around in it."
This comments allows no room for judging people based upon who their people are. However, this is exactly what Aunt Alexandra wants them to do. She speaks to Atticus to convince him that the children need to act like Finches, descendants of the man who founded Finch's Landing.
Aunt Alexandra's arrival has already been unsettling. In the South, Scout notes, someone coming to stay could mean a short time or a very, very long time. Up until this point, the children have been expected to be polite, but both are rough and tumble kids. They spend their days in the summer outside as long as possible. What Jem does, Scout does. She is a tomboy, never wearing dresses. Overalls are synonymous with Scout. Aunt Alexandra, however, wants they children —especially Scout—to act seemly or appropriate to their standing in society.
So Atticus tries to explain this to them. He speaks to his children about "gentle breeding" and living up to the family's name.
[Aunt Alexandra] asked me to tell you you must try to behave like the little lady and gentleman that you are. She wants to talk to you about the family and what it's meant to Maycomb County through the years, so you'll have some idea of who you are, so you might be moved to behave accordingly.
Atticus finishes talking "at a gallop," which infers that he is as uncomfortable speaking of this to them and they are in hearing it from him. The children are stunned, and it so goes against Atticus' grain, that he snaps at Scout. This is hard for the children to fathom, as they know who they are and it has nothing to do with their social standing.
Scout is beside herself:
My father never thought these things. My father never spoke so.
The children ask how they should do what has been asked of them for they wouldn't know where to begin. However, before Atticus leaves the room, he tells the children to forget all about it. In this we see that Scout's observance, "I knew he had come back to us," is accurate.
The conflict exists in two opposing perspectives: acting as society expects and dictates (which is Aunt Alexandra's belief) or being true to the person you are—which Atticus has taught his children is based on character alone.
The tone here (reflected in Atticus' advice and example) is that one's character amounts to everything important about a person; his or her standing in the community counts for nothing.