When Scout, Jem, and Dill are at the jailhouse, how does Jem react after Atticus tells him to go home and why?
In Chapter 15 of To Kill a Mockingbird, a crowd of men comes to the Maycomb jail, where Atticus is guarding Tom Robinson inside. They clearly plan to get Atticus out of the way and lynch Tom Robinson so that they can carry out vigilante justice and deny Tom his right to a trial, however flawed that trial might be. Atticus tells Jem to go home (and to take Scout with him) to protect him, but Jem stands his ground:
"As Atticus’s fists went to his hips, so did Jem’s, and as they faced each other I could see little resemblance between them: Jem’s soft brown hair and eyes, his oval face and snug-fitting ears were our mother’s, contrasting oddly with Atticus’s graying black hair and square-cut features, but they were somehow alike. Mutual defiance made them alike" (page 152; page numbers vary by edition).
Jem refuses to go home because he thinks the men might harm Atticus in his absence. He also refuses to go home because, as this passage implies, he is quite similar to Atticus. While Jem resembles his mother (who is deceased) in physical traits, his character is like that of Atticus. Like Atticus, Jem has a great deal of integrity, and once he decides something is right, he will not back down. Atticus is of course similar in his defense of Tom Robinson. It is, in the end, Scout who gets the men to back down when she reminds Mr. Cunningham that she is friends with his son, Walter, and she speaks about the entailment case in which Atticus is helping Mr. Cunningham. In response, Mr. Cunningham suddenly feels ashamed of acting as he has in front of children, and he tells the men to leave. As Jem and Scout are heading home, Atticus ruffles Jem's hair, which is a rare sign of affection. This gesture shows that Atticus understands that his son is a person of bravery and integrity.
Jem refuses to obey when Atticus tells him to go home. Atticus obviously is mindful of his children's safety at this point; he knows that the men have come to try and lynch Tom Robinson, and he does not want to risk his children being caught up in any violence. But Jem stoutly refuses to obey.
Jem does this because he (unlike the younger Scout) is also aware that violence might occur, that the men might hurt Atticus as Atticus is trying to protect Tom and Jem wants to help his father. From Atticus's point of view, it would be a bigger help if Jem went home to safety and took his little sister with him, but Jem wants to stay by Atticus's side.
Jem's refusal shows that he is being to assert his own individuality and manliness. He is on the verge of adolescence and is starting to make his own judgments and decisions. He still respects his father greatly, and likely will always do so. In the present incident, he defies parental authority not because he wants to be disobedient, but because he feels it's the right thing for him to do. He knows that Atticus, left alone to fight Tom Robinson's corner against the mob, needs support and he is not afraid to give it to him. This incident shows that Jem is cast very much in his father's mould; he too stands his ground in defence of what is right, and he shows himself to be physically and morally courageous like his father. Scout realises how alike they are in this moment as she looks from one to the other.
When the danger is over, Scout wrongly assumes that Atticus will get Jem into trouble for his disobedience. Instead, on the way home 'Atticus reached out and massaged Jem's hair, his one gesture of affection' (chapter 15). This shows that Atticus is proud of his son for supporting him at such a tense and difficult time.
How does Jem react when Atticus tells him to go home and why?
Jem refuses to go home, for several reasons. He wants to stand by his dad, and to protect him. He thinks doing so is right. He's growing up, and so rebelling against Atticus a bit more. And he's like Atticus, so it is in his character to stand firm when he thinks something is right.