We discover, after Mrs. Dubose dies, that she was a morphine addict that went clean in her last days. Atticus finds this incredibly brave. In fact, he calls her "the bravest person I never knew" because to go cold-turkey on such a hard habit, and to "leave this world beholden to nothing" was very difficult. Atticus admires that, even though she was "cantankerous". He is able to see past their disagreements to her true core of "real courage", which was when "you know you're licked before you begin but you begin anyway and you see it through no matter what." Because of her grit and determination to step into that impossibly difficult circumstance, she earned Atticus's respect.
When Mrs. Dubose requests that Jem come every day and read to her, Jem balks at this punishment for his cutting the camellia bushes after she says that Atticus is no better than the "trash he works for." Jem complains that the house has a gothic air: "There's shadows and things on the ceiling..." Atticus tells his son to pretend he is in Boo Radley's house, and he'll have fun.
Jem and Scout end up going to the house of Mrs. Dubose for a month; Jem reads from "Ivanhoe" while Mrs. Dubose corrects him, then slips into a daze in which she writhes and drools. Finally, after extended visits, Mrs. Dubose tells the children they do not need to come any more. Then, one night Atticus goes to visit Mrs. Dubose who has not been seen on her porch for a while. She has died, he tells the children; also, he explains the "fits" that Jem and Scout have observed: She was a morphine addict and withdrew with the help of the distraction of Jem's reading.
Atticus explains further that Mrs. Dubose could have remained on the morphine and not died in agony; however, she chose to meet death with a conscious mind--she would be "beholden to nothing and nobody." She has died "free." Atticus tells Jem that he wanted him to see something about Mrs. Dubose: real courage.
It's when you know you're licked before you begin, but you begin anyway and you see it through no matter what.
Of course, Atticus exemplifies this same courage in the courtroom at the trial of Tom Robinson as he does his best to defend his client, knowing that he is "licked before" he has started because of the prejudices of the jurors and town.