Atticus is older than other kids' parents, and Scout and Jem are sometimes embarrassed by their father's bookishness. When he gave Jem and Scout the air rifles they wanted for Christmas he didn't teach them how to shoot, instead only telling them not to shoot at mockingbirds, since it's a sin to kill a mockingbird. Miss Atkinson explains: all mockingbirds do is sing and create beauty and pleasure, so it's a sin to hurt them.
In a complicated world of good and evil, mockingbirds are one of the few things that are entirely good. The mockingbird, which gives the novel its title, is therefore a symbol of innocence and purity. Anyone in the novel who is purely innocent is a kind of mockingbird.
One day a rabid dog appears on the Finch's street. It's still far off, and Heck Tate, the sheriff of Maycomb, says only Atticus is marksmen enough to hit the dog from such a distance. Atticus kills the dog in one shot. Scout and Jem, astonished, learn that when Atticus was young he was the best shot in the county. Scout wants to brag at school, but Jem says not to: if Atticus was proud of it he would have told them.
Jem is growing up much more quickly now than Scout. He understands that honor and courage are about humility and service, not pride. Scout, meanwhile, just wants to brag.