The main conflict in this story is between past and present, between tradition and modernity.
The narrator faces this conflict in the form of her two daughters. Dee, the elder, has gone out into the world, acquired an education and certain, supposedly advanced ideas about what a modern African American woman should be like. Her other daughter, Maggie, lives quietly at home, following the old ways, doing the housework and preparing for marriage.
The conflict comes to a head in the shape of the old family quilts which Dee wants to take away with her to effectively preserve as museum pieces, quaint relics of a past which she feels she herself has left behind. However, the narrator has always intended these quilts for Maggie when she gets married. Maggie will actually use the quilts because to her and to the narrator they are part of an ongoing way of life, a living culture and not a dead past, as they are to Dee.
The narrator is forced to choose between the two when Dee implacably demands to be given the quilts. In an instant she makes her decision - indeed, it comes as a moment of revelation which imbues her with an almost religious ecstasy: 'just like when I'm in church and the spirit of God touches me and I get happy and shout'.
The narrator realises that the quilts properly belong to Maggie, to the traditional way of life; this is where they will be truly appreciated and used, and not flaunted as items of folk culture, as Dee would have it. Therefore she resolves the conflict by snatching the quilts from Dee and handing them to Maggie.
The main conflict presented in 'Everyday Use' is between dee and her family.