They're certainly entitled to think that, and they're entitled to full respect for their opinions... but before I can live with other folks I've got to live with myself. The one thing that doesn't abide by majority rule is a person's conscience.
This quote, said by Atticus, emphasizes one of the main themes of the book: the personal moral code versus social pressure. Atticus is a lawyer, so the majority rule is a thing he has to acknowledge and respect. But not only he is the skilled lawyer, he is also a man with a noble soul and he does his best to pass his principles to his kids. In the case of Tom Robinson these words are extremely relevant: Tom is black and Ewell’s daughter claims that she was disgraced by him - the two major triggers that cause the almost instinctive reaction of the society. The majority rule now works against Tom: almost everyone demands him to be executed. But Atticus risks his reputation and respect of his neighbors going against the majority and doing his best to really solve the case and help Tom clear his name.
Atticus’ conscience doesn’t allow him to just let the situation go the least troublesome way: he must stay defiant to the end, protecting Tom. This decision later almost costs him the lives of his children, when enraged Ewell tries to kill Jem and Scout to have his revenge. Ironically, the only thing that saves them is also connected with the words of the quote. Scout’s conscience didn’t allow her to despise Boo Radley as the majority of adults and some children did. She and Jem were kind to him and he repaid the kindness by saving their lives. Despite Jem’s skeptical attitude, Scout, who wholeheartedly believed in her father’s words, appears to be right. Sometimes not abiding by majority rule can help much more - at least help to stay a decent person in your own eyes and at most to stay alive.