In "Everyday Use" the primary conflict is between Dee and her mother (the first person narrator) and her sister Maggie. The mother and Maggie are on the same side of the issue, Dee opposes them.
The issue of who gets the quilts (treasured family heirlooms) is the focus of the conflict. Dee is a person who is used to getting her own way and bossing her family around. Maggie and the mother usually buckle under to the intense Dee, who is not very tolerant of their lifestyle.
When Dee essentially demands to take the quilts for herself, Maggie relents, as she always does, even though she was planning to take them with her when she got married. The mother finally gets up her courage and stands up to Dee, snatching the quilts away from her. Dee feels that her mother and sister have not kept up with the times as African-Americans, saying at the end, “You ought to try to make something of yourself too, Maggie. It’s really a new day for us. But from the way you and Mama still live, you’d never know it.”