I believe the first reference to the mockingbird comes in Chapter 10 when Atticus, who has just presented his children with air rifles for their Christmas presents, warns Jem to
"Shoot all the blue jays you want, if you can hit 'em, but remember, it's a sin to kill a mockingbird."
Later in Chapter 10, Scout discusses her father's statement with Miss Maudie, who tells Scout that
"Mockingbirds don't do one thing but make music for us to enjoy... That's why it's a sin to kill a mockingbird."
The newspaper editor, Mr. B. B. Underwood, refers to birds in his editorial following the death of Tom Robinson in Chapter 25.
He likened Tom's death to the senseless slaughter of songbirds by hunters and children...
Finally, in Chapter 30, Scout recognizes the symbolism of the mockingbird in the simple nature of Boo Radley, who has just saved her life from the attack by Bob Ewell. She knows that Mr. Tate "was right" in calling Ewell's death an accident, since exposing Boo to the "spotlight" of a murder investigation would
"... be sort of like shootin' a mockingbird, wouldn't it?"
Yet another songbird is also mentioned often in the novel: the finch.
The word "mockingbird" appears six times in the novel, and the word "songbird" appears once. In Chapter 10, Atticus tells Scout and Jem, "it's a sin to kill a mockingbird" (page 93). He cautions them against using a gun to shoot innocent creatures like mockingbirds. Miss Maudie repeats this on the same page and repeats the word "mockingbird" twice. Later in Chapter 10, as Atticus is chasing the mad dog, there is the following description: "the trees were still, the mockingbirds were silent" (page 98). In Chapter 21, Scout describes the courtroom during the Tom Robinson trial in the following way: "The feeling grew until the atmosphere in the courtroom was exactly the same as a cold February morning, when the mockingbirds were still" (page 214). In other words, everyone is waiting for the verdict in the trial, and it's as still and quiet as a morning without any songbirds singing. Later, at the end of Chapter 30, when Atticus asks Scout if she understands why Mr. Tate claims that Bob Ewell fell on his knife, she says she does and says, “Well, it’d be sort of like shootin‘ a mockingbird, wouldn’t it?” (page 280). By this, Scout means that prosecuting Boo Radley for killing Bob Ewell would be like going after an innocent creature like a mockingbird. In Chapter 25, Mr. Underwood writes an editorial in which he "likened Tom’s death to the senseless slaughter of songbirds by hunters and children" (page 244).
Mockingbirds and songbirds are used throughout the novel as symbols of innocence. They are also used to describe Tom Robinson and Boo Radley, who are innocent people persecuted by others around them.