Aunt Alexandra is a fanatic on the subject of her family's history, and it is important to her that Jem and Scout know about their heritage as well. Since the Finch family is one of the oldest families in the area, she also considers her past relatives among the elite of Maycomb. Simply put, in Alexandra's mind, the Finches are better than the other people of the area.
She never let a chance escape her to point out the shortcomings of other tribal groups to the greater glory of our own... (Chapter 13)
Other families have detrimental "Streaks"--"a Drinking Streak, a Gambling Streak, a Mean Streak, a Funny Streak"--but not the Finches. Aunt Alexandra considers the Finches "Fine Folks," which according to Scout, her aunt defines as
... the longer a family had been squatting on one patch of land the finer it was. (Chapter 13)
The Finches are also products of "gentle breeding," and that means Scout's tomboyish and unladylike ways conflict with Alexandra's idea of how young girls should behave. Although Alexandra has her reasons for stressing family heritage,
I never understood her preoccupation with heredity. Somewhere, I had received the impression that Fine Folks were people who did the best they could with the sense they had." (Chapter 13)