Atticus shoots the mad dog in chapter 10 and surprises his children, who were under the impression that he was a talentless, boring person. Before the drama unfolds, Jem and Scout lament that Atticus has no special talent and is too old to do anything interesting or fun. Later on, the children spot a rabid dog walking down the street and Calpurnia notifies the authorities. Sheriff Tate and Atticus arrive on the scene, and the sheriff looks to Atticus for help. Because of their negative perception of Atticus, Jem and Scout are astonished when Sheriff Tate hands Atticus his rifle, and Atticus kills the rabid dog in one shot. Scout describes the dramatic moment by saying,

With movements so swift they seemed simultaneous, Atticus’s hand yanked a balltipped lever as he brought the gun to his shoulder. The rifle cracked. Tim Johnson leaped, flopped over and crumpled on the sidewalk in a brown-and-white heap. He didn’t know what hit him (Lee, 88).

After killing the rabid dog, Atticus picks up his glasses, and Jem is paralyzed by the spectacle. The Finch children find it hard to comprehend what they witnessed, and Jem is awestruck by his father's extraordinary talent. Once Atticus leaves the scene with Sheriff Tate, Miss Maudie provides insight into Atticus's outlook on life and describes his hidden talent by telling the children,

If your father’s anything, he’s civilized in his heart. Marksmanship’s a gift of God, a talent—oh, you have to practice to make it perfect, but shootin’s different from playing the piano or the like. I think maybe he put his gun down when he realized that God had given him an unfair advantage over most living things. I guess he decided he wouldn’t shoot till he had to, and he had to today (Lee, 90).

When Scout wonders why Atticus is not proud of his exemplary shooting abilities, Miss Maudie replies, "People in their right minds never take pride in their talents" (Lee, 90). Following the incident, Scout declares that she cannot wait to brag about her father's special talent at school, but Jem encourages her to be humble by not saying anything about it.

It should be noted that different editions of To Kill a Mockingbird will have these quotes on different pages, but if you look for them in chapter 10 you should be able to find them.

In chapter 10, Jem spots a mad dog named Tim Johnson staggering down the main street of Maycomb. After informing Calpurnia, she recognizes that the dog has rabies and calls the operator to relay the message to the entire community that they should lock their doors and stay inside. Shortly after Calpurnia calls the operator, Sheriff Tate arrives with Atticus. Sheriff Tate knows that Atticus is the best shot in Maycomb and gives Atticus his rifle to shoot the mad dog. Atticus reluctantly takes Sheriff Tate's rifle, and Scout describes the scene by saying,

"In a fog, Jem and I watched our father take the gun and walk out into the middle of the street. He walked quickly, but I thought he moved like an underwater swimmer: time had slowed to a nauseating crawl...Atticus pushed his glasses to his forehead; they slipped down, and he dropped them in the street. In the silence, I heard them crack. Atticus rubbed his eyes and chin; we saw him blink hard" (Lee, 99).

After witnessing their father kill the rabid dog in one shot, both Scout and Jem are amazed and shocked at their father's marksmanship abilities. Later on, Miss Maudie explains to the children that Atticus's nickname is One-Shot Finch. 

The scene of Atticus shooting the mad dog, Tim Johnson, in Chapter 10 has ramifications in other chapters of To Kill a Mockingbird. It initially serves to show Jem and Scout that their father is not just "old" and "feeble," but that he has a special skill that none of their other classmates' fathers have: He was once the best marksman in Maycomb County--a skill about which he has never told his children.

     In a fog, Jem and I watched our father take the gun and walk out into the middle of the street...
     The rifle cracked. Tim Johnson leaped, flopped over and crumpled on the sidewalk... He didn't know what hit him.

Through Miss Maudie's explanation, the children also discover about humility. When Scout wonders aloud why Atticus wouldn't have told them about his shooting skill, Maudie tells her that

     "People in their right minds never take pride in their talents..."

Jem is so proud of his father that he exclaims,

"Atticus is a gentleman, just like me!"

In the following chapter, Atticus reminds Jem that Mrs. Dubose was a true example of courage--"the bravest person I ever knew." He points out the difference between her own style of courage and that of a man with a rifle.

"I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand."

Much later, as the jury returns to deliver its guilty verdict in the Tom Robinson trial, Scout pictures her father again, and

... it was like watching Atticus walk into the street, raise a rifle to his shoulder and pull the trigger, but watching all the time knowing that the gun was empty.