A mockingbird is an innocent creature that is targeted without deserving it.
When Atticus gives his children guns for Christmas, he tells them not to shoot at mockingbirds. This becomes a metaphor for Scout. Symbolically, a mockingbird does no harm.
Miss Maudie explains why Atticus tells his children not to shoot at mockingbirds.
"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)
There are two symbolic mockingbirds in the story: Boo Radley and Tom Robinson. Boo Radley is actually identified as a mockingbird by Scout at the end of the story. Scout tells Atticus that she understands why Heck Tate decided to say Bob Ewell fell on his knife, rather than identify Boo Radley as his killer—even though he was saving the children’s lives.
"Well, it'd be sort of like shootin' a mockingbird, wouldn't it?" (ch 30)
Scout describes Boo as a mockingbird because he does no harm, and has been only a joy in the children’s lives. He is targeted by society for no reason other than that he is different. Similarly, Tom Robinson is targeted because of his race.