There are four major types of conflict in a story: character versus self, character versus character, character versus society, and character versus nature. In this story, one might argue that the main conflict exists between Della Dillingham and herself (character vs. self) because she agonizes over what to get for her husband, Jim, for Christmas, as well as how to pay for it. She's only been able to save $1.87, and she is heartbroken that she will not be able to get him the kind of gift she feels he deserves: something of which he can be proud. Della wrestles a bit with herself, hence the conflict, eventually deciding to sell her hair in order to have enough money for such a gift.
One might also argue that the main conflict takes place between Della and Jim (character vs. character). Remember that conflict doesn't necessarily mean that one side is good, a hero, and one side is evil, a villain. To be an antagonist in literature means that one is an opposing force, an instrument of plot development, and perhaps an agent of change on the part of the protagonist. In the end, both Della and Jim have sold their most prized possessions in order to purchase something nice for the other, and those sales have rendered the gifts they receive essentially unusable.