Like many contemporaneous Southern parents, Atticus has given air-rifles to his children. But Atticus, being Atticus, wants Jem and Scout to use their guns responsibly. The last thing he wants is for them to start going round indiscriminately shooting innocent creatures such as mockingbirds.
Mockingbirds are completely harmless animals; they never cause anyone any bother. They just sit in the trees all day, sweetly singing their song. That's why it would be a sin to kill one of them, as Miss Maudie helpfully explains.
As the story develops we begin to realize that as well as the literal mockingbirds who live in the trees, Maycomb also has more than its fair share of figurative mockingbirds. There are many innocent souls who never do anyone any harm, but are nonetheless marginalized and despised by society for all kinds of different reasons.
The main case of the metaphorical mockingjay would be Tom Robinson, an African-American tried and convicted on a trumped-up charge of raping a white woman for no other reason than racism.
Then there's Arthur "Boo" Radley, who is stigmatized by the community as some kind of boogeyman. His reputation sticks despite the fact that there's no hard evidence that he's ever harmed a living soul. All kinds of urban legends have accrued to his name over the years, but there's nothing to suggest that he, like Tom Robinson, is anything other than one of life's mockingbirds.