Atticus tells Jem and Scout his definition of real courage in chapter 11, soon after Mrs. Dubose dies. Jem had just recently finished with his punishment of having to read to her because of his having destroyed her camellia bushes, and he is still feeling rather resentful toward her. When she has Atticus give him a gift-wrapped camellia, Jem at first thinks she is somehow mocking him, but Atticus helps him to see her gift differently.
As they soon learn from Atticus, Mrs. Dubose had just died - "cantankerous to the end" (148). She had spent her final weeks in great pain as she tried to break her long-term addiction to morphine. It was for that reason, Jem and Scout learn, she wanted them to read to her. They provided a distraction for her as she waited a bit longer each day to take her medicine, going through physical withdrawals in the process; finally, after much suffering, she was "free" of her addition.
After telling them this, Atticus says he would have made them read to her even if Jem hadn't destroyed the flowers:
"I wanted you to see what real courage is, instead of getting the idea courage is a man with a gun in his hand. It's when you know you're licked before you begin, but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew" (Lee 149).
He knows the kids greatly admired him for being able to shoot the mad dog in chapter 10, but that's not the kind of thing he wants them to look up to. If they are going to respect him for anything, he'd rather it be for taking on the Tom Robinson case - where, he knows he's licked before he begins, but he's going to begin anyway and see it through no matter what because it's the right thing to do.
That's true courage - it goes far beyond a simple physical act; it shows a deep moral integrity and strength, and a willingness to do what is right even though it may be hard, even though it may lose you friends, even though it may cause others to chastise or ridicule you, and even though it may cause you physical pain and suffering.