In a word, she learned that Atticus was a great father. 

As Atticus speaks with Uncle Jack during the time frame that the children are supposed to be in bed, he makes some very intentional statements because he knows Scout is listening in. He tells Jack that he hopes both Jem and Scout will not be affected by the social aspects of his defense of Tom Robinson. He shows that he values both of his children, and after he and Jack have the conversation about his defense, he tells Scout to return to bed. At the end of this chapter, Scout, our narrator, tells us that she realized years later that Atticus was saying all that he said with the intent of her hearing it.

Above all else, Atticus is a realist. He is not angry with Jack for spanking Scout, because he knows "she earned it." He understands that the case he has accepted--defending a black man who has been accused of raping a white woman--is a losing proposition. He realizes that he and his family will make enemies by defending Tom Robinson, and he knows that he has no chance of winning.

"The jury couldn't possibly be expected to take Tom Robinson's word against the Ewells'."

But Atticus' conscience guides him in making the decision, primarily because he could not "face my children otherwise." Atticus allows Scout to deliberately overhear them talking so that she knows the facts of the case; so that she knows how much he cares for and respects his children; and so that she can wonder what mysterious power Atticus employs to know that she was eavesdropping.

In Chapter 9, Scout loses her temper and ends up punching her cousin, Francis Hancock, in the face after he calls Atticus a "nigger-lover." Uncle Jack quickly intervenes during the scuffle and spanks Scout for her violent behavior. As soon as Scout gets home, she runs to her room, and Uncle Jack follows her. Uncle Jack ends up having a conversation with Scout and mentions that he was disappointed that she attacked Francis. Scout then proceeds to teach her uncle a lesson in how to treat and understand children. Scout mentions that Uncle Jack never gave her a chance to tell her side of the story which wasn't fair. She then explains to Uncle Jack that she felt extremely provoked when Francis called Atticus a "nigger-lover." Uncle Jack is shocked and feels terrible for not listening to Scout's side of the story. Uncle Jack then apologizes and begins to bandage Scout's bloody knuckle. While he is applying the bandage, Scout asks Uncle Jack, "What's a whore lady?" (Lee 54). Uncle Jack avoids telling her the true definition, and when he mentions it to Atticus, Atticus criticizes his brother for not being honest with Scout. Atticus tells Uncle Jack,

"When a child asks you something, answer him, for goodness' sake. But don't make a production out of it. Children are children, but they can spot an evasion quicker than adults, and evasion simply muddles em'. No" (Lee 55). 

Uncle Jack learns two important lessons about children at the end of chapter 8 in To Kill a Mockingbird.  The first is taught to him by Scout; the second, by Atticus.

The first lesson comes about as a result of Uncle Jack breaking up a fight between Scout and her cousin, Francis.  Scout pummels Francis after he calls Atticus a "Nigger-lover" (Lee 88).  When Aunt Alexandra and Uncle Jack break up the fight, he asks "Who started this?" and Francis points at Scout and says, "She called me a whore lady and jumped on me" (89).  Jack immediately punishes Scout, who shouts back that she will never speak to him again.

When Jack comes to her room later that night to try to make amends, Scout explains why she got so angry:

"You ain't fair ... you ain't fair.... In the first place you never stopped to gimme a chance to tell you my side of it - you just lit right into me.  When Jem an' I fuss Atticus doesn't ever just listen to Jem's side of it, he hears mine too, an' in the second place you told me never to use words like that except in ex-extreme provocation, and Francis provocated me enough to knock his block off -" (90).

Scout goes on to explain what Francis had said about Atticus, and Jack quickly determines that Francis is the one who should be punished.  Scout, however, tells him to let it go, asks him not to tell Atticus what she has said, and changes the subject by asking him what a "whore-lady" is.  To this, Jack responds with a long winded answer that completely avoids the question.

This is where the second lesson comes in.  When Jack tells Atticus about Scout's question and his response, Atticus says:

"You've a lot to learn, Jack.... When a child asks you something, answer him, for goodness' sake.  But don't make a production of it.  Children are children, but they can spot an evasion quicker than adults, and evasion simply muddles 'em" (92).

Once again, Atticus shows his wisdom in how he approaches raising his children.  He is frank, forthright, and respectful of both their feelings and intelligence.  Jack, chastised, is left to ponder the lessons he has learned.

In Chapter 9, Scout mentions that every Christmas Uncle Jack Finch comes to visit and stays a week along with Aunt Alexandra and her grandson Francis, whom she finds tiresome. This year Uncle Jack has two long packages for his nephew and his niece. Scout remarks that she finds her uncle a bit odd:

Jem and I always thought it funny when Uncle Jack pecked Atticus on the cheek; they were the only two men we ever saw kiss each other.

But, Scout adds that he is one of a few men of science who do not frighten her because Uncle Jack does not act like a doctor. As proof, she relates the time that she had a splinter in her foot, but Uncle Jack had her laughing so hard that she did not feel him pull it out.

Uncle Jack is a confirmed bachelor; the "only woman [he] can stand" is his cat Rose Aylmer. Later, when Scout uses some expletives of which Jack does not approve, he chastises her after dinner as she sits upon his knee. But, he learns from Scout that "bad things" are being said about Atticus, and Francis is one of those participating. Later, Scout overhears Uncle Jack asking Atticus about the case.

One thing Scout learns is that adults don't have all the answers. In Jack's conversation with Atticus, he tells his brother that he actually learned a few things from Scout about how he could have handled the conversation better. Thus, she also learns that her uncle is willing to admit he's made a mistake when the situation warrants doing so. And her uncle remains true to his word, not telling Atticus the real reason Scout fought with Francis, just as Scout had requested. Therefore, she learns that she can trust Uncle Jack, even if he did jump to conclusions and punish her unfairly earlier. This goes a long way in repairing any tears in the seams of their relationship.

Scout also learns a few things about her father. She learns that Atticus is aware of the impending conflict Scout might face because of his role as Tom's defense attorney, and he is trying to guide his children through this tough situation by teaching them the proper ways they should respond to the certain conflict which awaits them. She also learns that her father thinks that the principles of this trial are important enough that he's willing to risk exposing his children to some temporary societal discomforts so that they don't catch "Maycomb's usual disease."

In short, Scout learns that both her uncle and her father are men of principle and can be depended upon for wisdom in tough situations—even if, like Uncle Jack, they falter from time to time.

From overhearing her father and Uncle Jack discussing Atticus's having accepted the role of defense attorney in the case against Tom Robinson, Scout truly comprehends the meaning of Miss Maudie's description of her father as "...the same inside his house as he is on the public streets." Also, she learns of "Maycomb's usual disease" and how her father wants his children to ignore the gossip.

In Chapter 9 Scout listens from around the corner as Atticus talks to his younger brother, who asks Atticus if he could not get out of taking the Robinson case. Atticus replies that he really has no choice: "...do you think I could face my children otherwise?" For, if he refused the case when it was given to him by Judge Taylor, Atticus would have displayed hypocrisy as he has always expressed the idea that everyone should be treated fairly.

Then, Jack asks of this forthcoming trial, "...how bad is this going to be?" and Atticus responds,

"It couldn't be worse, Jack. The only thing we've got is a black man's word against the Ewells'....The evidence boils down to you-did--I-didn't. The jury couldn't possibly be expected to take Tom Robinson's word against the Ewells'...."

Atticus expresses his despair of winning, but he believes that he will have "a reasonable chance" on appeal. A little later, Atticus voices his wish that Jem and Scout will not listen to the townspeople's gossip about this trial. In addition, he hopes that they will come to him and trust him.

In To Kill a Mockingbird, by Harper Lee, the family goes to visit Aunt Alexandra for Christmas. While they are there, news about Atticus taking on Tom Robinson's case comes up. (Though Aunt Alexandra is very conservative and would not bring it up with the children around, Scout's cousin Francis does.)

Francis proceeds to taunt Scout about her father's part in defending a black man. Scout, not one to mince words, calls Francis a "whore lady" and she goes after Francis until she finally gets him.

When stalking one's prey, it is best to take one's time. Say nothing, and as sure as eggs he will become curious and emerge.

When Francis comes out, Scout does her best to convince him that she is no longer mad, and then pops him in the mouth.

Scout and her Uncle Jack had had a serious discussion about her language. He felt Scout was not behaving in a ladylike fashion, and warned her that the next time she spoke inappropriately, without "extreme provocation," he would punish her.

When Francis repeats what Scout called him, Uncle Jack spanks her.

I found myself suddenly looking at a tiny ant struggling with a bread crumb in the grass.

Scout promises Uncle Jack she will never speak to him again, and Atticus decides to pack the children up and go home. When they arrive at the house, Scout immediately runs to her room. Soon Uncle Jack comes to speak with her. First she gets sassy, but when he threatens go spank her again, she stops and listens to him.

When Uncle Jack tries to tell her that she had the punishment coming, she asks to explain herself and tells Jack that she loves him, but he doesn't know how to deal with kids. When he puzzles over this, she explains.

When Jem and Scout fight, Atticus always allows both to tell their side of the story. Jack did not do so. Scout also complains stating that she was provoked because of the names Francis called her father; she refused to let him insult Atticus that way and retaliated.

Jack is furious at Francis and wants to see him punished, but Scout tells him to forget—she doesn't want Atticus to know why they were fighting. She also reminds Jack that Francis did not get away with it; obviously she is satisfied with hitting him in the mouth, and she asks Uncle Jack to bandage her hand.

Uncle Jack has to learn to listen when a child tries to tell him something important without jumping to conclusions: he needs to hear "both sides."

In chapter 9, Atticus takes Jem and Scout to visit Aunt Alexandra, Frances her son, and Uncle Jack. Frances antagonizes Scout into a fight by first talking bad about Dill and then going further to insult Atticus. He informs Scout that the family looks down on Atticus. Frances says:

Grandma says it's bad enough [Atticus] 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'.

This infuriates Scout and she chases him, eventually punching him in the mouth. Uncle Jack stops Scout before she can do more damage. Upon finding out that Scout has used foul language, he proceeds to spank her. Later, Uncle Jack tries to explain to Scout why she deserved a spanking and Scout tells him that he wasn't being fair and that he doesn't "understand children much." When he allows her to explain, she tells him that he didn't give her a chance to tell her side of the story, and that "Atticus doesn't just listen to Jem's side of it, he hears [hers] too." She also said that he had told her never to use foul language "except in ex-extreme provocation, and Frances provocated [her] enough to knock his block off-"

Later, when Uncle Jack talks to Atticus about the incident, he says that Scout doesn't know the meaning to half the bad language she uses. Uncle Jack tells Atticus that he avoided answering her question when she asked what a "whore-lady" was. Atticus immediately corrects him and tells him,

When a child asks you something, answer him for goodness' sake. But don't make a production of it. Children are children, but they can spot an evasion quicker than adults, and evasion simply muddles 'em...Bad language is a stage all children go through, and it dies with time when they learn they're not attracting attention with it.

By telling Uncle Jack this, Atticus is merely letting him know that it is best to be honest because if he's not, then children won't trust him. And trust is a big part of Atticus' parenting style.

The scenes with Uncle Jack are actually in Chapter 9. After Cousin Francis twice calls Attiucs a "nigger-lover," Scout gets even with him when she "split my knuckle to the bone on his front teeth." Uncle Jack apparently spanks Scout--she has "seven or eight red marks" afterward--and Scout swears she'd never speak to him again. But she relents, the two talk, and Uncle Jack discovers the reason for Scout's attack on her cousin. Scout scolds him for not allowing her to explain before he spanked her. Jack later tells Atticus of their discussion, and he claims that Scout was right.

    "I shall never marry, Atticus."
    "Why?"
    "I might have children."

Atticus then scolds his brother for not being straight with Scout.

   "Jack! When a child asks you something, answer him for goodness' sake... children can spot an evasion quicker than adults and evasion simply muddles 'em."

Jack learns that he has a lot to learn about handling children.

While visiting the Finches, Uncle Jack learns to take Scout's arguments seriously.  It is by now common knowledge that Atticus is defending a black man, which despite the respect people have for Atticus, is looked down upon.  Scout's cousin Francis visits and runs around the house taunting Scout about her father's job and saying that Atticus has become a " n*****-lover.”  Scout punches Francis and is pulled off Francis by Uncle Jack, who scolds her.

Scout is not apologetic about her actions.  While Uncle Jack at first seems preoccupied with scolding Scout, Scout lashes out at him and says that he never listens to her side of the story.  Uncle Jack is taken aback, and does stop to consider Scout's perspective.  He apologizes to Scout, and learns to respect her side of the story.