Discussing both characters is too much for one question. Please ask for an explanation of the other character in a different question.
When Harper Lee’s novel To Kill a Mockingbird begins, Jem Finch is ten years old. Scout begins by mentioning the ending of the story for Jem who breaks his arm when he is thirteen. Although he sometimes teases and aggravates Scout, Jem becomes a good brother. He protects her, serves as one of her best friends, and in the end, saves her life. Many things impact Jem between that beginning sentence and the actual event.
The three years that are covered by the story are a transitional period for Jem as for any boy. He enters puberty and begins adolescence. That is enough in itself to be difficult without the life altering events that Jem experiences. The best gauge of Jem’s maturity comes from the development of his courage.
Initially, Jem loves to play the childhood games fun to all children. The children especially enjoy the fantastic creations about Boo Radley. Jem and Scout seem to understand Maycomb and its inhabitants, particularly Boo.
When Dill begins to come and visit during the summers, the children become obsessive about getting Boo to come outside. In the beginning, Jem’s bravery extends to touching the side of the Radley’s house and running.
As the story progresses, Jem begins to change. He becomes more pensive and quiet. Scout calls her brother moody. There is more to it than that because Jem acquires the qualities that will fill out his character when he becomes an adult. He feels compassion toward Boo. When the hole in the tree is blocked, Jem cries. He is able to see the bias and prejudice that fill the hearts and minds of some of the people in his town.
Finding his place in the adult world, Jem tries to teach Scout how to behave with Aunt Alexandra. When Dill runs away, Jem tells Atticus to prevent anything from happening. It is obvious that Jem wants to be as much like his father as possible.
One of the biggest changes in Jem issues forth from the trial of Tom Robinson. As Jem listens to the adult discussions of the trial, it is evident that Jem has the ability to understand the difficult issues that are involved in the legal case. He is astounded when the guilty verdict is given.
His [Jem’s] face was streaked with angry tears as we make our way through the cheerful crowd. “It aint’ right," he muttered all the way to the corner of the square where we found Atticus waiting...
“It ain’t right, Atticus,” said Jem.
No son, it’s not right.” We walked home.
His admiration for his father grows as he understands that his father has the courage to stand up for what he believes. Miss Maudie understands the hurt that Jem feels after the trial is over and helps Jem when she gives him a piece of the adult cake.
Unknown to the children is the revenge that Bob Ewell has promised to extract from Atticus and his children. Atticus had embarrassed and practically accused Ewell of the crime for which Robinson had to pay.
The night of the pageant shows how courageous Jem has become. When he hears or feels that someone is following them, Jem worries about Scout. He protects her by fighting the shadowy figure with no concern for himself.
It is in this fight that Jem has his arm broken by Bob Ewell; however, he saves Scout from the initial attack by Bob. Of course, Boo saves the day by killing Ewell and carrying Jem home to his father.