Scout begins her narration by talking about Jem's broken arm. She then goes on to suggest that the events that led up to Jem's accident had been instigated by the Ewell's while Jem believed the original event leading up to the accident had been when Dill came to visit them for the first time. Scout then goes on to say it went further back and gives a brief history of how her family came to Maycomb and the current state of affairs there.
Scout discusses her family life. She lives with her father, Atticus; her brother, Jem; and their cook, Calpurnia (and effectively, a mother figure in the home). Scout adds that her mother died when she was two so she has little memory of her. As Scout relates all of this, she goes back to the summer that Dill first came to visit; this is the beginning of the story that led up to Jem's aforementioned accident/broken arm.
The remainder of the chapter concerns Jem, Scout, and Dill getting to know each other. Dill is intrigued by the mystery of Boo Radley. Scout narrates some of the gossip (some from Stephanie Crawford) about the Radley's. Then, the three children discuss ways of getting a chance to see Boo. This establishes the mystery of Boo Radley, something that will occupy the children's time periodically throughout the novel. The chapter ends with Jem taking a dare, from Dill, to touch the Radley house.
Adding on to amarang's answer. Here on enotes.com we have the summary for chapter 1, and also the rest of the book! You should check the "To Kill a Mockingbird" Study Guide!