reminder that we need to sympathize with and understand all people, even those who seem detestable at first.

Mrs. Dubose is so unlikeable that Jem destroys all of her prized camellia bushes. To punish him, Atticus makes Jem read to her every day for a month. Mrs. Dubose's house is truly horrible: "An oppressive odor met us when we crossed the threshold, an odor I had met many times in rain-rotted gray houses where there are coal-oil lamps, water dippers, and unbleached domestic sheets" (Chapter 11; page numbers vary according to the edition of the book). Mrs. Dubose is similarly repulsive, as she has saliva on her mouth and continually berates Jem and Scout because Atticus is defending Tom Robinson.

After the month of Jem's reading to Mrs. Dubose is over, Jem and Scout feel relieved. Shortly thereafter, Atticus tells them that she died. She had been addicted to morphine, a pain killer, but was able to kick her addiction before she died. Atticus says, "She said she was going to leave this world beholden to nothing and nobody...She meant to break herself of it before she died, and that's what she did" (Chapter 11). In other words, she could have easily died still addicted to morphine, but she wanted to show that she was stronger than the drug. Jem was reading to her to help her pass the time and forget about her pain.

Mrs. Dubose's character is a testament to the idea that everyone has secret struggles. Atticus asks Jem and Scout to treat Mrs. Dubose with empathy, even though she at first seems dislikable. As Atticus tells his children at the beginning of the book, we can never really understand someone until we walk around in their shoes, and Mrs. Dubose is an example of an unlikeable person who we sympathize with once we understand her inner pain. 

We meet Mrs. Henry Lafayette Dubose in chapter eleven of Harper Lee's To Kill a Mockingbird, and it is not a particularly pleasant meeting. The old woman is cranky and cruel as she sits on her porch and viciously insults all of the Finches every time they pass by her. 

Of course we know that Jem loses his temper one day and damages some of Mrs. Dubose's flowers. As a punishment, Jem has to go and read to the old lady, and that is not a pleasant task, either. Later we learn that the old woman was addicted to morphine as a painkiller and determined to break free of the addiction before she died.

The answer to your question about why the author included Mrs. Dubose in the novel comes from Atticus as he is talking to Jem after Mrs. Dubose died. It is interesting to note that this chapter on Mrs. Dubose follows the chapter in which Atticus demonstrated courage by facing down a rabid dog in the middle of the street. While this is one kind of courage, Atticus wants his children, and Jem in particular, to understand about other kinds of courage. Atticus says:

"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.”

Harper Lee includes this episode with Mrs. Dubose as part of an ongoing theme expressed in the novel, again through Atticus. In chapter three, he explains to Scout that

"[y]ou never really understand a person until you consider things from his point of view […] until you climb into his skin and walk around in it."

Jem's encounter with Mrs. Dubose is designed to teach him this same lesson. There was a reason for the woman's crankiness and bad behavior, something neither Jem nor Scout could see but which Atticus was able to understand--and even overlook--because he saw things from Mrs. Dubose's point of view. Though it is not clear that Scout understood all of this, Jem is moved by the lesson.

Miss Dubose is in the story to demonstrate the theme that real courage is fighting even if one knows the battle is impossible or highly improbable to win.  She is dying but is determined to break her addiction to the pain killer that was given to her as a treatment for the pain the disease caused.  She didn't want to die with something she considered a sign of weakness.  This is the same sort of courage it took for Atticus to take on the Tom Robinson case.  He knew it was a case he couldn't win with a jury in Maycomb, but he gave it his best shot.  He fought knowing he'd lose in the end, just as Miss Dubose fought her addiction knowing she'd die anyway.

Mrs. Dubose serves as model of two concepts: courage and discipline.

Mrs. Dubose's problem of addiction elicits her effort to free herself. As an old woman who is taking morphine for some type of pain, we know that she has reason to be cranky as readers. The children however, think she is just mean. If she is in continual conflict with her body but trying to die free from addiction she has to practice both courage and discipline. It takes bravery to to allow her own body to endure the pain when she knows there is a quick fix for the pain if she chooses to use it. Additionally it takes discipline, the kind of discipline that is the practice of doing something right over and over, to be able to achieve longer and longer lengths between her doses of morphine.

The practice of that actually disciplines Jem as well because he has to read to her for longer and longer each time. Jem deserved discipline as in a punishment for his crime of destroying her flowers.

Teaching courage and discipline were both necessary because of what Atticus and the children were about to endure from the town. It helped teach the children a lesson about patience and keeping their cool when other people say terrible things that just demonstrate heated emotions.