In Harper Lee's novel, the mockingbird symbolizes a certain innocence or, more accurately, a lack of guilt and blameworthiness (which is not exactly the same as innocence).
The lesson of the mockingbird is that the vices of cruelty and aggression are evident when people attack those who do not deserve to be attacked.
Miss Maudie explains the mockingbird's significance at one point.
"Mockingbirds don’t do one thing except make music for us to enjoy. They don’t eat up people’s gardens, don’t nest in corn cribs, they don’t do one thing but sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.”
The mockingbird has a beauty of its own and an autonomy that should be respected. To harm a living creature that is itself doing no harm is a sin.
Cruelty, in the novel, is the result of a lack of empathy. Scout learns this lessons in a number of places in the novel but nowhere more directly than in a discussion she has with her father.
“First of all,” he said, “if you can learn a simple trick, Scout, you’ll get along a lot better with all kinds of folks. You never really understand a person until you consider things from his point of view—“
“—until you climb into his skin and walk around in it.”
Rushing to judgement, especially judgement based on rumors or bias can lead a person to commit the sin of attacking those who are not deserving of attack.
The characters that become "mockingbirds" in the novel include Boo Radley and Walter Cunningham, two figures subjected to unwarranted criticism and unfounded condemnation (to greater and lesser degrees, respectively).
"Tom Robinson himself is another “mockingbird.” Innocent of the crime for which he is tried, he has to face the evil of hatred simply because of the color of his skin" (eNotes).
Each of these characters is wronged and is judged without consideration of any his actual merits. Gossip, superficial class biases and wrong-minded race prejudice are some of the contributing factors to the false views that arise around these characters.
Thus the title of the novel comes to reinforce the lesson of the mockingbird offered in the novel. Practice empathy. Understand others before judging them. Do not harm those who have done no harm. Question your biases.
By challenging his children to learn to be empathetic and to base their opinions on actual experience instead of hearsay, Atticus demonstrates a compassionate ethos that inspires his children to keep an open mind about the world they live in (and the people they share it with).