A symbol in the novel and what its importance to understanding character or theme?
The most obvious symbol in the novel is related to the title: a mockingbird. Theme and character are closely interwoven with the title, as you can tell from studying the novel's major protagonists.
In Chapter 10, Atticus tells Jem that he can shoot all the bluejays he wants, but "...it's a sin to kill a mockingbird" (Lee). Miss Maudie explains that while blue jays are pests, (eating people's gardens and making nests in corn cribs), mockingbirds do nothing but "sing their hearts out" for people. They are innocent.
What characters in the novel are also innocent, and yet targeted? Think about Boo Radley, and the stories about him being a "malevolent phantom" and such early in the novel. Contrast those stories with Scout's encounter with Boo at the end of the novel.
And what about Tom Robinson? Clearly, he is the most important character symbolic of an innocent person killed--in this case literally killed--by unjust people. Even more minor characters in the novel such as Dolphus Raymond can be considered "mockingbird characters"--in that they are mistreated, ostracized, or marginalized unfairly, with often tragic consequences.