In Chapter 10 of To Kill a Mockingbird, author Harper Lee begins to connect her mockingbird symbol to social inequality through the concept of innocence.
In Chapter 10, both Jem and Scout are given air-rifles as gifts. Atticus, knowing that the kids would shoot at birds rather than tin cans, tells them its okay to shoot bluejays but warns them against shooting mockingbirds, saying, "it's a sin to kill a mockingbird." Miss Maudie explains later that all mockingbirds do is "make music for [people] to enjoy." In contrast, other birds make nuisances of themselves by eating up gardens and nesting where corn harvests are stored. Hence, in contrast to other birds, mockingbirds are both very different and very innocent.
The mockingbird symbol relates to racial and other social prejudices because society often judges those they deem to be socially "different" or "weaker" as also being morally inferior, even though they are actually innocent. For example, Tom Robinson is innocent of the crime he's charged with but is found guilty simply because of the color of his skin. Lee is using her mockingbird symbol to show just how sinful it is to hurt an innocent person and to misjudge that person simply because society sees that person as being "different."
We see the connection between the mockingbird symbol and social inequality solidified towards the end of the book when Scout connects having Arthur Radley charged with killing Bob Ewell to shooting a mockingbird. When Scout and Jem are attacked by Ewell, Radley runs to their defense and kills Ewell with the knife Ewell had intended to kill the children with. Sheriff Tate argues against charging Radley in favor of protecting him from the town's ridicules since he acted to save the children. Scout sees Radley's innocence and says to her father that charging Radley would "be sort of like shootin' a mockingbird" (Ch. 30). Hence, just as Scout sees towards the end, it is cruel to ridicule or torment those who are innocent simply because they are seen as being socially "different."