You will find the answer in chapter 11 of To Kill a Mockingbird. Jem and Scout have disliked Mrs. Dubose for many years. She is an old lady who lives a couple of houses away from them. If they speak to her, Mrs. Dubose usually yells back at them and criticizes them. She talks badly about Atticus taking on the case of Tom Robinson, and this makes Jem very angry. Jem pulls up some of her flowers, and for punishment he has to read to her. Scout goes along with him while he reads to her, not knowing the real reason as to why they are doing this. Atticus is fully aware of the reason, since he is Mrs. Dubose's attorney, but he doesn't tell Jem what the reason is because he wants Jem to learn a lesson. When Jem finds out that Mrs. Dubose has died, Atticus tells Jem a little about Mrs. Dubose. Jem learns that Mrs. Dubose was addicted to morphine for many years and wanted to kick the habit of it before she died.
"She said she was going to leave this world beholden to nothing and nobody. Jem, when you're sick as she was, it's all right to take anything to make it easier, but it wasn't all right for her. She said she meant to break herself of it before she died, and that's what she did."
Jem is surprised when Atticus calls Mrs. Dubose a lady. He wonders how Atticus could respect her when she talked the way she did about Atticus.
"She was. She had her own views about things, a lot different from mine, maybe...son, I told you that if you hadn't lost your head I'd have made you go read to her. I wanted you to see something about her- 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. 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."
Atticus is explaining why Mrs. Dubose was so brave to him, but he doesn't realize that he is also describing himself. Atticus knows that there is no way he will be able to win the case of Tom Robinson, but he gives it his all. That is what makes someone brave, and Atticus is brave.