In Chapter 11, what does Atticus imply when he says that Mrs.Dubose is a model of real courage rather than "a man with a gun in his hand"?
The implication of this quote is best understood when it is viewed in context of the previous chapter. The quote in question is from chapter 11; however, Atticus's quote is referencing an event that was described in chapter 10.
In chapter 10, Scout and Jem are outside playing, and they see off in the distance a dog acting a bit strangely. It turns out that the dog is Tim Johnson, and it has gone mad. Calpurnia rushes Jem and Scout inside, and she calls Atticus at his office. Atticus and Heck Tate arrive a bit later, and they confirm that the dog has rabies. The dog needs to be shot and killed before it harms anybody, but Tate isn't confident that he can make the shot. Tate hands his rifle to Atticus, which seems very odd to Jem and Scout. Atticus is averse to guns, and the kids don't know that he is a great shot. Atticus shoots and kills the dog from range. He doesn't have to get up close and personal with it. Because of the gun, Atticus isn't really ever in danger of being harmed by the dog. The gun allows him to be "brave" from a distance.
Atticus's quote in chapter 11 is how Atticus explains what real courage is. By mentioning the gun, he is pointing out to Jem and Scout that there is a very real difference between his courage in chapter 10 and Mrs. Dubose's courage.
"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. She broke her addiction to morphine, which was a very hard thing to do. She was the bravest person I ever knew.”
Atticus wants Jem and Scout to see that Mrs. Dubose chose pain and suffering. She chose to fight hard against an enemy that would win no matter what. Atticus's fight against the dog wasn't really a fight. Atticus knew he was going to win because he had the tools to do it; therefore, bravery and courage had little to do with his actions. Mrs. Dubose is a completely different story. She's guaranteed to lose her fight, and she has a way to lose painlessly. She can stay on the morphine; however, she doesn't choose that easy out. She faces her "enemy" head on and in her right mind. To Atticus, that is the bravest thing that anybody can do.
I also think that Atticus's message is trying to hint that his acceptance of the court case is a braver thing to do than shoot a gun and end a life. At this point in the story, Jem and Scout don't understand exactly what Atticus has been asked to do; however, Atticus knows that he has likely accepted a fight that he can't win. He's going to fight hard like Mrs. Dubose, but he is ultimately still going to lose.
Atticus regards Mrs. Dubose as a woman of courage because she lives life on her own terms. She became a morphine addict after a doctor prescribed it for pain years earlier. When the doctor informs Mrs. Debose she has a few months left to live, she decides she will not die as a morphine addict. She endures withdrawal and dying without the palliative care of morphine. As Atticus points out, Mrs. Debose died conscious and cankerous. Although Mrs. Debose criticized Atticus for defending a black man and condemned his children, he still admires her strength, conviction, and courage. A man with a gun can end conflict quickly from a distance, but Atticus realizes true bravery is found inside no matter how scary or uncomfortable the situation.