Miss Maudie is the Finches’ next-door neighbor and friend.
When Scout and Jem are young, she is just a neighbor. During the course of the events of the books she becomes a good friend, supportive and offering comfort and advice.
Jem and I had always enjoyed the free run of Miss Maudie's yard if we kept out of her azaleas, but our contact with her was not clearly defined. (ch 5, p. 29)
Scout spends time with Miss Maudie when Jem does not want to hang out with her, and whenever she needs company or advice. Miss Maudie is the mother figure, and a fellow girl.
She had never told on us, had never played cat-and-mouse with us, she was not at all interested in our private lives. She was our friend. (ch 5, p. 31)
Miss Maudie advises the children on Boo Radley, and a variety of other issues. She does not talk down to them or make fun of them. As Scout notes, she does not laugh at them unless they tell a joke they intend to be funny.
Miss Maudie is also a source of information about Atticus, since she has known him for a long time. When the children are dissatisfied with their father, she tells them some of his background, including the fact that he is a crack shot.
Miss Maudie’s wise advice is actually the source of the book’s title.
Mockingbirds don't do one thing but make music for us to enjoy. They don't eat up people's gardens, don't nest in corncribs, they don't do one thing but sing their hearts out for us. That's why it's a sin to kill a mockingbird. (ch 10, p. 64)
This quote is indicative of Miss Maudie’s outlook on life. Along with Atticus, she is one of the children’s moral teachers. She helps them see what is important in life, and how to treat people.
When the children are sad, they go to Miss Maudie for comfort. She often makes them little cakes. After the verdict, she comforts them in a way that demonstrates she knows Jem is growing up.
There was a big cake and two little ones on Miss Maudie's kitchen table. There should have been three little ones. … But we understood when she cut from the big cake and gave the slice to Jem. (ch 22, p. 151)
Miss Maudie is acknowledging that even though Jem is disappointed by the verdict, he is becoming an adult. He is growing up, while Scout and Dill remain children. This is an indicator both of Miss Maudie’s tact and her sensitivity to what is going on in the children’s lives.
Miss Maudie Atkinson is one of the relatively few ladies in Maycomb whom Jem and Scout find truly congenial. This has a lot to do with the fact that, like Atticus, she doesn't talk down to children, she treats them more as equals. This makes it easy for Scout and Jem to talk to her. She is a rather unconventional member of the community; for instance she likes to spend time in her garden in her overalls, unlike other ladies who dress femininely and stay in the house. She is straight-talking and fearless in her manner, which earns her a fair amount of respect in the commmunity even if she doesn't subscribe to the majority's views. She is a good influence on the children as she has a broad and tolerant outlook, like Atticus.