Mrs. Henry Lafayette Dubose, "the meanest old woman who ever lived," lives two doors up from the Finch family and loves to antagonize Scout and Jem whenever they pass by. After Mrs. Dubose insults Atticus, Jem attacks her camellia bushes with a baton; his punishment for this act is that he must go read to her every day for a month, a task which is perhaps made more bearable by the fact that Scout comes with him.
After twenty minutes of listening to Jem read Ivanhoe while Mrs. Dubose corrects his mistakes, Scout notices that Mrs. Dubose seems to fall into a strange, almost lifeless state, or "fit." Soon after, an alarm clock goes off, and Mrs. Dubose's servant sends the children home so that she may administer Mrs. Dubose's medicine. This pattern continues each day, with the period that Jem must read getting longer every time and the alarm clock going off later than usual.
Mrs. Dubose dies shortly after Jem's month of reading is completed, and Atticus reveals to the children that the woman had been a life-long morphine addict. She had wanted to "leave this world beholden to nothing and nobody," and so she lengthened the reading sessions to serve as a distraction from the pain of her morphine withdrawal. Ultimately, she was able to kick the habit and died as clean "[a]s the mountain air."
Mrs. Dubose is a morphine addict. She knows she is about to die. She wants to die free from addiction to that drug. However, because of her great pain, this is difficult to do. The kids coming to read to her is one way she can at the very least be distracted and hope that she can go longer in between doses. Most addicts know that this begins the process of weaning themselves off of their addiction, put more time in between consumption moments.
For the kids, this just seemed mean at the time. They didn't understand what she was really working towards. Once she does pass, Atticus reveals to them her great courage and what she was trying to do in the end of chapter 11.