right thing because it is the right thing to do, not because there is a reward or "win" at the end.

This connects to Atticus's courage in mounting a real defense of Tom Robinson even though he too "is licked" before he begins. Like Mrs. Dubose, he does the right thing, all the way through to the end.

Although the page number will vary (depending on the publisher and year of publication), we can be sure that the quote originates from chapter 11 of To Kill a Mockingbird. If you are looking for the quote in the book, it is located at the very end of chapter 11. The edition I am referring to is the 1982 paperback edition, published by Warner Books. It has 284 pages, and the quote in question is on page 115.

The quote is certainly a very important one. In it, Atticus explains that courage is a mental decision. Without mental and emotional fortitude, it is impossible to demonstrate courage when it is most needed. Atticus wanted Jem to distinguish between true courage and the outward appearance of courage. 

To Jem, Mrs. Dubose was a cantankerous old lady. She had a sharp tongue, which she often unleashed on those who offended her. On the surface, there was not much to like about Mrs. Dubose. It was only after her death that Jem learned about her heroic struggle with morphine addiction. Atticus tells Jem that Mrs. Dubose weaned herself from the drug and died a free woman. 

Atticus wanted Jem to learn that even the most controversial characters can surprise us. His words here echo an important quote from the last chapter of the book:

Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them. 

There are several mentions of courage in To Kill A Mockingbird, but the most significant conversation between Atticus and Jem occurs after Mrs. Dubose dies on pg. 149 at the end of chapter 11. Atticus states:

"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 was fighting a morphine addiction. She succeeded in stopping her morphine right before she died. While Jem was reading to her, she was taking less and less.

Atticus wants Jem to get to know Mrs. Dubose because she is demonstrating a different kind of courage. It’s not a physical kind of courage like facing down a mad dog. It’s a more important kind of courage- mental courage, and courage in the face of emotional and physical pain.

This quote comes at the end of Chapter Eleven. In the chapter, Atticus tells Jem the real reason he had him read to Mrs. Dubose. When the elderly woman was alive, she was a morphine addict. Towards the end of her life, she fought an excruciating mental and emotional battle to overcome her addiction. Because she died victorious in her quest, Mrs. Dubose will always be a 'great lady' in Atticus' mind.

Jem isn't too thrilled at Atticus' verdict of Mrs. Dubose's character; he still remembers all the vitriol the old lady hurled at Atticus when she was alive. However, Atticus counters with the above quote; he maintains that courage isn't always evident and overt. Someone with a gun isn't necessarily brave; intrinsically, courage is a mental decision unrelated to the amount of firepower at one's disposal. Mrs Dubose exhibited this truth magnificently when she fought courageously through overwhelming physical and mental pain to overcome her crippling addiction to a powerful drug.

The above quote is also significant in that it highlights Atticus' own courage when he later represents Tom Robinson, a black man, in court. Like Mrs. Dubose, Atticus lives the courage of his convictions without apology. Despite his community's anger and disapproval, he fights to acquit Tom. The above quote supports the theme of courage or grace under fire; both Mrs. Dubose and Atticus display great courage in spite of insurmountable challenges. They live 'beholden to nothing and nobody' and their lives are a demonstration of their true characters.

This refers both to Atticus's skill as a sharpshooter and Mrs. Dubose's struggles to overcome morphine addiction. Atticus is considered courageous for killing the rabid dog, but as he understands and tries to communicate to his children, his skill with a gun is an inborn talent that has nothing to do with real courage.

Mrs. Dubose, the subject of Atticus's quote, shows true courage in battling her addiction and coming clean before she dies. There's no reason for her to fight this battle, but she does it anyway because she wants to be true to herself. That, Atticus says, shows genuine mettle.

The quote also refer to him—though he is not thinking of himself at the time. He shows true courage and integrity in going against the racist customs of his town and mounting a true defense for Tom Robinson. He knows he can't win and that people will hate him for it, but he does it anyway because it is the right thing to do.

Atticus Finch displays integrity when he speaks of Mrs. Dubose in a laudatory manner to his son Jem.

Three synonyms for integrity are honesty, decency, and fairness. Atticus exhibits these three qualities when he speaks to his son after Mrs. Dubose's death. For he decently puts aside the differences in views that he and Mrs. Dubose have had on specific issues such as defending Tom Robinson. He has also forgiven her for having upbraided his children and vilified him. In fairness to Mrs. Dubose, Atticus tells Jem that Mrs. Dubose demonstrated "real courage" by taking herself off the morphine to which she was addicted because she wanted to die "beholden to nothing and nobody." When Atticus tells Jem, "She was the bravest person I ever knew," this is an honest compliment because he feels that Mrs. Dubose possessed real courage in going through painful drug withdrawal. For Atticus explains real courage as "when you know you're licked before you begin but you begin anyway, and you see it through no matter what" (Chapter 11). Mrs. Dubose was "licked" because she was in such poor condition; nevertheless, she withdrew from her morphine so that she could die free of the drug.

Atticus demonstrates his integrity without flaw, throughout To Kill a Mockingbird. In this quote from Chapter 11, he is talking about Mrs. Dubose. Despite the fact that she's cranky, generally unliked, and has even made racist comments about Atticus' involvement with Tom Robinson, Atticus considers what life must be like for her. He never assumes. And here, he doesn't simply assume that Mrs. Dubose is just a mean, hateful woman. For whatever reason, Atticus is privy to the information that Mrs. Dubose had been addicted to morphine (perhaps as a result of some illness, likely the one that killed her). Mrs. Dubose knew her death was approaching but she wanted to kick her morphine addiction before she died, to "die free" as Jem put it. 

Atticus notes that he and Mrs. Dubose did not agree on many issues, but she demonstrated courage in doing what she thought was right. Near the end of Chapter 11, Atticus says, "According to her views, she died beholden to nothing and nobody." 

This description of Mrs. Dubose is analogous to the trial. Atticus knew it would be difficult, if not impossible, to get Tom a verdict of "not guilty." But, he did his best anyway. This is how he demonstrates his integrity. He notes that courage is doing the right thing even in spite of the likelihood that you will fail and in spite of the possibility that no one will notice your efforts. This may sound redundant but what Atticus means is that courage is doing the right thing for no other reason than it is the right thing to do: not for reward or approval. 

This is spoken by Atticus to Jem.  In the previous chapter, Jem was amazed to find out that Atticus is an impressive shooter, the best in town.  This knowledge inspires respect in Jem.  However, Atticus doesn't want Jem thinking that courage is connected with having a gun and shooting things.  He doesn't want Jem to think that courage has to do with physical actions or fighting.  Atticus wants Jem to understand that standing up in the face of certain failure is what courage is.  If you know that you will fail, but you insist on standing up for what you believe in, or fighting for what is important to you, then you are showing real courage.

This is what Mrs. Dubose did.  She had the kids reading to her every day because she needed a distraction.  Her sickness caused her great pain, and she had become addicted to morphine.  She didn't want to be dependent on anything - she didn't want to be an addict.  Even though she knew she was about to die, Mrs. Dubose was determined that she would die free, not controlled by some substance.  This is the type of courage that Atticus was talking about.