After Aunt Alexandra made her temporary move back to Maycomb, she made herself right at home. Part of her grand design was to help Atticus bring up Jem and Scout in a more proper manner and, specifically, to oversee Scout's transition into a lady. After a dispute over the history of cousin Joshua St. Clair--Alexandra called him a "beautiful person," while Scout remembered that Atticus had said he "went round the bend" and tried to kill someone--Alexandra demanded that Atticus have a talk with Jem and Scout about the Finch heritage.
"... Your aunt has asked me to try and impress upon you and Jean Louise that you are not from run-of-the-mill people, that you are the product of several generations of 'gentle breeding.' "
Atticus seems to be uncomfortable with this term, but to appease his sister, he continues to explain that both of them would like Jem and Scout "to behave like the little lady and gentleman that you are." The "gentle breeding" simply means that the Finch's can boast a proud and honorable lineage: They are not white trash like the Ewells, or poor country folk like the Cunninghams. Their family was one of the founders of the area, and Alexandra is particularly proud of this fact. Atticus reassures Scout that things will not be different between them, and jokes that "I get more like Cousin Joshua every day, don't I?"
I know now what he was trying to do, but Atticus was only a man. It takes a woman to do that kind of work.