Atticus is able to find good in nearly all humans (except for possibly Bob Ewell), even in those whose positive traits are rarely exhibited. Such is the case with Mrs. Henry Lafayette Dubose. Angry, mean-spirited and anti-social, Mrs. Dubose nevertheless impresses Atticus with her courage and inner strength. He explains to Jem why she is such a "great lady."
"... 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 you 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... She was the bravest person I ever knew."
Atticus knew that Mrs. Dubose's morphine addiction is what caused her to be so irritable toward everyone around her. He also knew the difficulty in going cold turkey in order to become drug-free before she died. She died bravely and on her own terms--two examples of courage that impressed Atticus greatly.
Mrs. Dubose, one of the Finch's neighbors, did not approve of Scout and Jem, she did not approve of the way Atticus was raising them, and she did not approve of Atticus' defending Tom Robinson. In spite of her continual nastiness to the children, Atticus taught them to respect her:
"Easy does it, son," Atticus would say. "She's an old lady and she's ill. You just hold your head high and be a gentleman. Whatever she says to you, it's your job not to let her make you mad."
It takes more moral courage to hold one's head high with someone like Mrs. Dubose than to sass her back. However, when she insulted Atticus, Jem cut down all of her camellias, totally ignoring his father's advice.
When Atticus finds about the destruction of the flowers, he is dismayed, telling Jem:
". . .but to do something like this to a sick old lady is inexcusable"
Atticus forces Jem to read to Mrs. Dubose, day after day, as retribution for having destroyed her flowers. This is a totally odious task to Jem. Unknown to the children until afterwards, Mrs. Dubose was addicted to pain medication - morphine. After she dies, Atticus tells Jem that his reading to her helped her overcome her addiction through force of will. It is clear that Atticus admires her moral courage.
"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."
Also, with regard to his defending Tom Robinson, Atticus explains to the children that he could not live with himself if he did not defend Tom, he felt it was his moral duty. He says:
"...but before I can live with other folks I've got to live with myself. The one thing that doesn't abide by majority rule is a person's conscience."
This shows that he has moral courage himself.
You can read all of these things in chapter 11. See the analysis here on eNotes.