In chapter nine of To Kill a Mockingbird, Atticus's brother, Uncle Jack, comes to stay with the Finch family for Christmas. The family goes to visit Uncle Jimmy, Aunt Alexandra (Atticus's sister), and cousin Francis at the Landing. Francis proves to be an immediate nuisance, and Scout considers him to be "the most boring child [she] ever met." He is a tattletale who reports back to Aunt Alexandra, who in turn nags Atticus about his parenting abilities.
At the dinner table, Francis irritates Scout by teasing her about Dill, who he refers to as a "little runt," and tells Scout that she is "mighty dumb sometimes." He then reveals the startling attitude of Aunt Alexandra, stating:
If Uncle Atticus lets you run around with stray dogs, that's his own business, like Grandma says, so it ain't your fault. I guess it ain't your fault if Uncle Atticus is...
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a nigger-lover besides, but I'm here to tell you it certainly does mortify the rest of the family--
This revelation--that Aunt Alexandra is judgmental enough to refer to Atticus in such a way--is deeply disturbing to Scout. Scout goes on the offensive, attacking Francis and chasing him off into the kitchen, eventually splitting her "knuckle to the bone on his front teeth." Unfortunately for her, Uncle Jack intervenes quickly and punishes Scout with a "licking."
In chapter 9, the Finch family gets together to celebrate Christmas, and Scout is forced to spend time with her boring, antagonistic cousin, Francis Baker. After dinner, Scout and Francis go outside, and Francis begins criticizing Atticus for defending a black man and not raising his children properly. Francis repeats Aunt Alexandra's derogatory comments, which reveal that she is a racist. Francis tells Scout,
Grandma [Aunt Alexandra] says it’s bad enough he lets you all run wild, but now he’s turned out a nigger-lover we’ll never be able to walk the streets of Maycomb agin. He’s ruinin’ the family, that’s what he’s doin’ (Lee, 85).
Aunt Alexandra clearly does not support her brother's defense of Tom Robinson and is ashamed that he is willing to defend a black man. Alexandra's racist comments indicate that she subscribes to the prejudiced ideologies of Maycomb's community. Later in the novel, Aunt Alexandra expresses her concerns about Atticus defending Tom Robinson but sympathizes with Atticus. She witnesses firsthand how the trial negatively affects Atticus and wishes that he would simply not defend Tom Robinson.
I assume that you are talking about what Francis says to Scout in Chapter 9 when the clan is all at Finch's Landing. The things Francis says to Scout while he is making fun of her make it quite clear that his grandmother (Scout's Aunt Alexandra) is racist and also very arrogant and stuck up.
Francis tells Scout about Alexandra's attitude towards Dill. She clearly looks down on Dill because of his family situation. She compares him to a stray dog. Aunt Alexandra has also, apparently, been saying what they thought was bad things about her brother Atticus. She has told Francis that Atticus is a "nigger-lover."
By saying those things to Francis, she has shown that she thinks she is better than others because of her family background and she has shown that she is racist.
How does Francis reveal Aunt Alexandra's unpleasant traits in To Kill a Mockingbird?
Just as Aunt Alexandra gossips about the different families of the area, she apparently gossips inside her family. While Francis and Scout eat their Christmas dinner at Finch's landing, he tells her that his grandmother [Alexandra] is going to teach him to cook. Scout giggles and remarks that she would not want Dill waiting on her; further, she reveals that she is going to marry Dill.
Francis hooted.
"You mean that little runt Grandma say stays with Miss Rachel every summer?...Grandma says he hasn't got a home...he just gets passed around from relative to relative...."
Francis also reveals that Aunt Alexandra disapproves of Atticus's allowing Scout to "run around with stray dogs[Dill],....and he adds that he guesses it also is not her fault that "Uncle Atticus is a "n****r-lover besides." And, he adds, his grandmother says that they will "never be able to walk the streets of Maycomb" because Atticus is ruining the family.
These cruel remarks of Francis that are meant to insult her cause Scout to attack her cousin in defense of her father and her friend Dill. Of course, Scout also sees Aunt Alexandra in a new light.