In Chapter 10, Atticus tells Scout and Jem that he'd rather they shoot tin cans than birds with their air-rifles, but if they are going to shoot birds, to shoot bluejays instead of mockingbirds. When Scout asks Miss Maudie about what Atticus has said, Miss Maudie explains that mockingbirds don't do anything but sing. They make no mischief, such as eating people's gardens or nesting in corncribs, so it's a sin to kill them because all they do is make music for people to enjoy.
The mockingbird symbolizes pure innocence. It stands for characters in the novel such as Tom Robinson and Boo Radley who are entirely innocent and who do not hurt anyone. However, Tom and Boo are vilified because they are different than those who hold power in Maycomb--Tom because he is black and Boo because he is a social outsider. For that reason, these innocent people are victimized as if they were innocent mockingbirds that were shot while singing in a tree.
Atticus tells the children it is a sin to kill a mockingbird because "they don't do one thing but sing their hearts out for us." Mockingbirds are not harmful so it would be wrong to hurt them. The mockingbird is symbolic of Tom Robinson and Boo Radley. Both characters never harm anyone. In fact, they both keep to themselves, only interacting with others in the form of help. Tom helps Mayella. Boo helps the children most significantly when he saves them from Mr. Ewell. The mockingbird, and these two characters, symbolize innocence and generosity. It is prejudice that clouds many of the Maycomb residents' judgment. They can't see that Tom and Boo are mockingbirds in this symbolic sense because their preconceived notions about the two of them are founded on racism and gossip. It is fitting at the end of the novel when Atticus agrees with Scout that most people are nice "when you finally see them."