Jem is a very important part of the narrator Scout’s life. In fact, the first example of his character is revealed in the very first paragraph. We know about Jem before we know much about Scout.
When he was nearly thirteen, my brother Jem got his arm badly broken at the elbow. When it healed, and Jem's fears of never being able to play football were assuaged, he was seldom self-conscious about his injury. (ch 1)
This information tells us quite a bit about Jem. First of all, we know that he is not self-conscious. He is an athlete. He takes things in stride, meaning he does not get easily put off track by things.
Jem is four years older than Scout. This means that while he is a playmate, he often serves a parental role too. Scout sometimes resents being compared to Jem, so we know he is well-behaved. For example, Calpurnia seems to favor Jem.
She was always ordering me out of the kitchen, asking me why I couldn't behave as well as Jem when she knew he was older, and calling me home when I wasn't ready to come. (ch 1)
Yet we also learn that Jem is not always a perfect angel. He can be disobedient, such as in playing the Radley game when Atticus has told them not to.
For example, Jem gets angry and destroys Mrs. Dubose’s flowers when she continually insults them about their father.
He did not begin to calm down until he had cut the tops off every camellia bush Mrs. Dubose owned, until the ground was littered with green buds and leaves. (ch 11)
This is an example of how Jem’s laid-back nature has its limits.
Jem also shows signs of courage and integrity. When he loses his pants on Boo Radley’s porch, he returns to get them.
When he shows 'em to Atticus it'll be pretty bad, that's all there is to it. Go'n back to bed."
"That's what I know," said Jem. "That's why I'm goin' after 'em." (ch 6)
Scout is sure he’ll be shot, but Jem shows bravery and goes anyway. He is more concerned with making sure that Atticus trusts and respects him than the physical danger. This shows that Jem is maturing.