At the beginning of Chapter 10, Scout discusses how she and Jem wish Atticus was not so old. She adds that he doesn't do anything exciting, no hunting or fishing, and that he is older than the fathers of other children her age. Scout recalls the time Atticus gave them air rifles but would not teach them how to shoot. Atticus does tell them to shoot at tin cans and blue jays but not to shoot mockingbirds because it is a sin to do so.
Scout asks Miss Maudie why this is a sin and she replies:
Mockingbirds don’t do one thing but make music for us to enjoy. They don’t eat up people’s gardens, don’t nest in corncribs, they don’t do one thing but sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.
This is where the title of the book comes from. It is a sin to kill something/someone who does no harm to anyone and only helps and/or sings for others.
Later in this chapter, Atticus must show his shooting skills as he shoots and kills a rabid dog, Tim Johnson. Scout and Jem admire him for this and Jem respects the fact that Atticus doesn't brag about his skill.
In Chapter 11, after some indignant and rude responses from Mrs. Dubose, Jem secretly cuts the buds off of her camellia bushes. Atticus finds out and demands that Jem talk to Mrs. Dubose about it. She wants Jem to read to her six days a week for a month. Jem complies and at the end of the month Mrs. Dubose dies. Atticus reveals that she was very sick and at the last stages of her life, she decided to kick her morphine habit. She'd sent Jem a perfect camellia in a box shortly before she died. Jem is confused and overwhelmed. Atticus concludes by telling Jem that despite their differences, Mrs. Dubose was the bravest person he ever knew.