Atticus Finch brought it up first. He said it was a sin to kill a mockingbird.
"'Remember it's a sin to kill a mockingbird.' That was the only time I ever heard Atticus say it was a sin to do something, and I asked Miss Maudie about it."
Miss Maudie gives the best definition of a mockingbird:
“Mockingbirds don’t do one thing but . . . sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.”
Tom Robinson is a mockingbird in that he doesn't do one thing wrong. All he does is provide pleasure to the folks he encounters. That is exactly how he got in trouble. Tom Robinson was helping Mayella with some chores. He was humming a melody as he chopped up the dresser drawers. He was giving of himself charitably. He was in no way harming anyone.
Then when Bob Ewell came home and found Tom Robinson, he set out to ruin his life. Tom was innocently helping Mayella when Bob Ewell accused him of raping his daughter. The white community assumed Tom was guilty, thus killing Tom with their conviction before he even had a fair trial:
Tom’s goodness is recognized by very few of the characters in the book. His willingness to help Mayella with the many tasks she has to perform around the house is seen by the white community as inappropriate. They can not understand why a black man would feel pity or compassion towards a white woman. They themselves would not feel the urge to help an Ewell, the most despised of the white population and so, to them, the only reason for Tom to offer such help would be for a sinister purpose.
Boo Radley is considered a mockingbird because he had no evil intentions against anyone. Boo was misjudged. Jem and Scout judged him based on rumors. Boo is not evil. He helps Jem and Scout out many times. First, he left gifts in the knothole of the tree. He has a warm expression of his affection for Jem and Scout. He sewed Jem's pants when Jem got caught in the fence and had to come out of his pants.
Boo Radley puts a blanket on Scout the night Miss Maudie's house burns down. Most importantly, Boo saves Jem's and Scout's lives. On the night Bob Ewell attacks Jem and Scout, Boo kills Bob Ewell in his effort to protect Jem and Scout from Ewell's evil attack.
Tom Robinson and Boo Radley never had intentions to hurt anyone. They are caregivers. They provide beautiful music for those they try to help. They never did anything but sing their hearts out through their good deeds. They were protectors of others. Both Tom and Boo are misjudged based on evil rumors. It is a sin to to kill Tom Robinson but he dies.
Boo Radley cannot live a life of privacy due to the evil rumors that circulate about him. It is a sin to treat him with such disrespect.