I'll put the most important quotes in bold text below for easy reference.
In the beginning of the novel, Scout only mentions Mrs. Dubose a little bit here and there, talking about what a mean old lady she is, how she finds fault with Scout and Jem and everything they do, and so on. The kids clearly hate her. But Atticus wants them to be kind, to show understanding toward Mrs. Dubose.
In Chapter 11, here's a brief conversation Atticus has with Jem:
“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.” Jem would say she must not be very sick, she hollered so.
As you can see, Atticus is telling Jem to cool it: to recognize that Mrs. Dubose is elderly and in bad health, and that he should take the high road and not show anger to her, no matter what she says. Jem resists this advice, telling his father that the old lady can't be that sick, since she has the strength to yell so much.
Later in that chapter, after Jem has gotten angry and destroyed Mrs. Dubose's flowers, Atticus chides him:
“Son, I have no doubt that you’ve been annoyed by your contemporaries about me lawing for n******, as you say, but to do something like this to a sick old lady is inexcusable. I strongly advise you to go down and have a talk with Mrs. Dubose,” said Atticus. “Come straight home afterward.”
Here, Atticus is telling Jem that although Mrs. Dubose made Jem angry with her comments, Jem shouldn't have damaged her plants, and now he has to go back to her house and apologize to her.
Jem begins reading to Mrs. Dubose after that, and he discusses what this is like with Atticus, who reminds Jem to be tactful and to overlook any of the lady's bad habits:
“Did she frighten you?” asked Atticus.
“No sir,” said Jem, “but she’s so nasty. She has fits or somethin‘. She spits a lot.”
“She can’t help that. When people are sick they don’t look nice sometimes.”
Finally, the whole last scene in Chapter 11 is worth rereading. This takes place right after Mrs. Dubose dies. Atticus talks to Jem about her again, revealing that she had been addicted to a painkiller, and that her fits that Jem had witnessed were the lady's brave efforts to end her own addiction. Here are some of the most memorable things Atticus says then about Mrs. Dubose to Jem:
"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."
"You know, she was a great lady."
"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."