Aunt Alexandra has a variety of beliefs about breeding and family.
1. All families have "streaks" in them. This means that there are common features that are passed down regardless of choice from generation to generation.
Aunt Alexandra, in underlining the moral of young Sam Merriweather's suicide, said it was caused by a morbid streak in the family. Let a sixteen-year-old girl giggle in the choir and Aunty would say, "It just goes to show you, all the Penfield women are flighty." Everybody in Maycomb, it seemed, had a Streak: a Drinking Streak, a Gambling Streak, a Mean Streak, a Funny Streak.
2. The Finches were high society people. The reason for their ability to be in the upper class had to do with the amount of time they had lived in the area:
Somewhere, I had received the impression that Fine Folks were people who did the best they could with the sense they had, but Aunt Alexandra was of the opinion, obliquely expressed, that the longer a family had been squatting on one patch of land the finer it was.
3. Gentle breeding was necessary for Jem and Scout. To Alexandra, gentle breeding meant that their mother was carefully selected to ensure that the Finches continue being fine folks. Her suggestions seemed to imply that love isn't necessarily important, class is. Atticus tries to explain this to the kids.
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' gentle breeding --'
"Gentle breeding," he continued, "and that you should try to live up to your name--' Atticus persevered in spite of us: "She asked me to tell you you must try to behave like the little lady and gentleman that you are. She wants to talk to you about the family and what it's meant to Maycomb County through the years, so you'll have some idea of who you are, so you might be moved to behave accordingly."
Obviously Alexandra was disappointed in the products that Jem and Scout were turning out to be for the family. This gentle breeding included having very special manners and behaviors that were required for high society Southern people.