The first example of what Atticus means by this comes up in the context of this statement. Mr. Tate, the sheriff, has come over to tell Atticus that Bob Ewell was killed falling on his knife. Atticus debates this version of events, saying that Jem stabbed him and that this has to come out in the open so people aren't whispering about it. When Atticus says he has to be the same inside the house and in town, he means that he can't say to the world at large he lives with honesty and integrity and then do something underhanded, like lie, to get his son out of a manslaughter or murder charge.

Atticus says:

if I connived at something like this, frankly I couldn’t meet his eye, and the day I can’t do that I’ll know I’ve lost him. I don’t want to lose him and Scout, because they’re all I’ve got.

Another example of Atticus needing to behave with integrity towards his children is when he tries to go along with Aunt Alexandra's ideas. He tells them they have to respect their family lineage. They shut down on him because they know he doesn't mean what he says. Finally, he admits to them that he doesn't care about a person's family history but only about the character of the person in question.

Atticus is referring to the way he lives his life. He's explaining that it would be hypocritical to behave one way in public and a different way at home with his children. He means that he cannot be racist or unethical in his dealings with people and then go home and tell his children not to be racists or cheats. Atticus believes all people deserve to be treated fairly and with respect. Atticus treats the poor with the same respect that he treats the wealthy, and more importantly, he treats the black people of Maycomb with the same kind of respect that he has for the whites. This is the reason why he feels he must defend Tom Robinson. Atticus truly believes in equality during a time when it wasn't popular or accepted by society. He doesn't expect his children to be more or less than he is himself. Atticus knows that children will emulate the actions of their parents, and he takes his job as a parent and as a role model very seriously.

I already mentioned Tom Robinson as an example of how Atticus treats people, and Walter Cunningham is another example. When Walter comes home with Scout for lunch, Atticus treats him as he would any guest in his home. When Scout makes fun of Walter, she is punished by Atticus for showing disrespect to a guest in their home. Atticus has a number of clients who cannot afford to pay him money, so many of them pay him with whatever they're able to give him, such as wood for heat.