"Shoot all the bluejays you want, if you can hit 'em, but remember it's a sin to kill a mockingbird."
The example of justice here is that since mockingbirds don't do anything wrong, it would be inherently wrong to kill one of them, and thus unjust.
"You felt sorry for her, you felt sorry for her?"
"Yes, suh. I felt right sorry for her, she seemed to try more'n the rest of 'em-"
This scene's example of justice is a little more obscure. Atticus is Tom's lawyer, and thus could either be his saving grace or his angel of death. Atticus is an expert rifleman, as displayed in this scene, and could thus easily shoot a mockingbird out of a tree, just as easily as he could let Tom Robinson die.
In this scene, the sense of justice is warped and twisted so that it is rather injustice. Miss Caroline reprimands Scout for simply trying to assist Walter Cunningham and explain the situation to her. Though the punishment is less than useless on Scout, the unjust punishment bestowed upon poor Tom Robinson leads to his death.
This scene of injustice is quite possibly the most obvious one in the whole book. Apart from Tom's sentencing, that is. This poor man who just wanted to help a lonely young woman from time to time is now being reprimanded for his compassion simply because he is black. Because he is black he must not feel bad for white folk, because he also has a hard time, and black people simply lack basic human traits, such as empathy, according to these people.
Tom Robinson is a mockingbird due to the compassion he shows to Mayella Ewell, metaphorically his mockingbird song, and he is literally and figuratively shot for it. He just wants to help her and this is what he gets for it.
Boo is a mockingbird due to his compassion toward the children. His song is both his gifts to them and his act of saving them from Bob Ewell. Heck Tate saying that Bob Ewell fell on his own knife is him attempting to save Boo from being "shot" AKA being overwhelmed with popularity, which, as a man who has been inside for many years, would be just about as bad as dying.