It is a sin to kill a mockingbird because they do not do anything to hurt anyone, they just sing.
When Atticus tells them that it is a sin to kill a mockingbird, Scout is surprised.
That was the only time I ever heard Atticus say it was a sin to do something, and I asked Miss Maudie about it. (ch 10)
Atticus does not use the word “sin” lightly, clearly. He does not have a problem with the children killing blue jays. Clearly blue jays are less than mockingbirds in his esteem. Scout wants to know why.
"Mockingbirds don't do one thing but make music for us to enjoy. They don't eat up people's gardens, don't nest in corncribs, they don't do one thing but sing their hearts out for us. That's why it's a sin to kill a mockingbird." (ch 10)
Mockingbirds are innocent creatures who do not hurt anyone. As opposed to blue jays, who bully other birds, mockingbirds just copy other birds and sing. They entertain the people, so the people should not hurt them. Atticus makes a point to mention mockingbirds because they are different.
The mockingbirds are symbolic. They are targeted because they are different. The difference does not have to be a bad thing. In this case, it is actually a good thing. Characters like Boo Radley and Tom Robinson are mockingbirds because they are innocents who are targeted because of their differences. Just as it is a sin to kill a mockingbird, targeting them because they are different is wrong.