The first sentence of To Kill a Mockingbird answers this question, but the entire novel elaborates on that answer. Scout, the narrator, begins the story by stating that her brother, Jem, "got his arm badly broken at the elbow." She goes on to explain that the break left a permanent disability, so that Jem's left arm was shorter than his right and hung abnormally at his side with the thumb perpendicular to his leg. The disability didn't affect the arm's function, and Jem was able to play football like the other boys.
As they aged, the two siblings looked back on the events that led to Jem's broken arm. Jem believed the starting point of it all was their attempt as children to draw their mysterious neighbor, Boo Radley, out from his house. The reason the stories about Boo are so important in the lives of the Finch children is that, if it hadn't been for Boo, Jem wouldn't have ended up with merely a broken arm. He very likely would have died that night instead.
After the Halloween program at school, Jem and Scout walk home through the woods in the dark. Bob Ewell assaults them. He has threatened revenge on Atticus Finch for defending Tom Robinson and for revealing in court that he, Ewell, was the one who beat Mayella, not Robinson. Although Robinson was unjustly convicted of rape, after the trial the townsfolk treated Ewell with even more contempt than they had before. On the night of the Halloween program, Ewell was drunk and wielding a knife as he followed the children in the woods; he might have murdered both children if Boo Radley had not been there to protect them.
So although the book begins by referring to Jem getting his arm broken, this turns out to be an understatement. What really happened to Jem was that he was assaulted and was the victim of a murder attempt.