In To Kill a Mockingbird, how and when does Jem Finch show that he is understanding?

Jem is a sensitive and understanding young man. Although he is only twelve years old when the novel ends, he is intelligent and precocious. Jem and Scout do not always get along, but he is always complex and is a good big brother. 

Jem is a good brother most of the time, and a good friend. He is compassionate toward Dill, the fatherless boy who comes to Maycomb every summer. Dill is younger than Jem, but Jem still enjoys his company. An example of Jem being understanding is when Dill is self-conscious about his father when Scout asks about him. 

I asked Dill where his father was: “You ain’t said anything about him.”

“I haven’t got one.”

“Is he dead?”

“No. . . ”

“Then if he’s not dead you’ve got one, haven’t you?”

Dill blushed and Jem told me to hush, a sure sign that Dill had been studied and found acceptable (Chapter 1).

Dill doesn’t talk about his father because he is long gone. It is a sore spot with Dill. He longs for a regular family. It is something he thinks he will have when his mother remarries, only to find his new stepfather ignores him. 

Another example of Jem’s maturity and compassion is his reaction to Boo Radley. At first, he finds the legend of their reclusive neighbor as a way to pass time. The more interactions the children have with Boo, though, the more Jem realizes Boo is a sensitive and caring individual. When Boo places a blanket on Scout’s shoulders during Miss Maudie’s fire, Jem tries to make sure Atticus does not tell Nathan Radley because he doesn’t want Boo to get in trouble. Jem says,

Mr. Nathan put cement in that tree, Atticus, an‘ he did it to stop us findin’ things—he’s crazy, I reckon, like they say, but Atticus, I swear to God he ain’t ever harmed us, he ain’t ever hurt us, he coulda cut my throat from ear to ear that night but he tried to mend my pants instead. . . he ain’t ever hurt us, Atticus— (Chapter 8)

Since Nathan Radley put cement in the tree hollow where Boo was leaving them presents, Jem is pretty sure Nathan will not understand Boo’s relationship with the kids. He knows Nathan Radley would not approve of the blanket. In telling Atticus to keep the blanket a secret, Jem is showing empathy for Boo.

Jem is particularly affected by the Tom Robinson trial. He sees the case based on facts, not skin color and social conventions. He is therefore horrified when there is a guilty verdict.

It was Jem’s turn to cry. His face was streaked with angry tears as we made our way through the cheerful crowd. “It ain’t right,” he muttered, all the way to the corner of the square where we found Atticus waiting (Chapter 22).

Jem was convinced that his father built his case. He thinks his father proved Tom Robinson could not have physically committed the crime. There was no rape. Mayella's father beat her because he found her with a black man. Jem is horrified and saddened by the injustice. It shows he is naive, but also that he is compassionate. He understands how the world should work.