Neighbor Miss Maudie Atkinson serves as a sort of advisor to Scout in To Kill a Mockingbird. She dispatches words of wisdom and sometimes explains to the children the reasons behind some of Atticus' decisions. She is also a friend and confidante to Scout, who knows she can trust Maudie.
... Jem and I had considerable faith in Miss Maudie. She had never told on us, had never played can-and-mouse with us, she was not at all interested in our private lives. She was our friend.
Miss Maudie (who is actually not a "Miss" at all, but a widow, just like Atticus) is down-to-earth and usually says what's on her mind, not unlike Scout's father. She enjoys baking cakes for the children and allows them the freedom to play in her yard. Miss Maudie and Atticus are also great friends, and his rare humor comes to life around Maudie more than anyone else. Atticus, of course, has a different relationship with Scout: He is the father and he must make the important decisions involving his children. Maudie merely serves as someone to lean on when she needs an older friend.
I think of Miss Maudie Atkinson as the female Atticus. Sometimes authors even give us clues that they intend for connections with character names. Look at how both names start with A-T... could be nothing... could be something.
Atticus generally gives his children his expectations verbally as if they are adults, and explains himself when necessary. We see this in his encouragements not to mess with the Radleys and his explanations of concepts like rape.
Maudie likewise confirms for the children right and wrong and enforces a respectful attitude of their father. She says he lets them win at checkers and that he does the right thing on behalf of the town when the town is too scared to act.
Scout's friendship with Maudie hinges on the fact that Maudie is the mother version of Atticus. Although she doesn't punish like a mother, she respects and listens to Maudie like a mother.
Both Atticus and Miss Maudie are authoritarian figures - they are most interested in the children behaving well in public. However, Miss Maudie also has conveys a warmer, more humanitarian persona. She takes the time to explain why the children (especially Scout) should behave the way she suggests; whereas Atticus is more of a top-down manager (do this because I say so and you know I am right). Miss Maudie is more respectful of the children and their ability to make the right decisions on their own, if given back-ground infomation.