This is a great question. In order to answer your question, let me make three points.
First, the conflict is that the two protagonists, Della and Jim, are poor and it is Christmas. In other words, they want to give a gift to each other, but they cannot afford anything. We get a sense of their poverty in the opening lines when, Della, complains that she only has "One dollar and eighty-seven cents."
Second, as both Della and Jim take assessment of what they can do, they realize that they only have two possessions that are worth anything. Della has her beautiful hair and Jim has a gold pocket watch. They decide to sacrifice these and sell them and with that money buy each other a gift. The irony is that Jim sells his watch to buy Della combs for her hair, and Della sell her hair to buy a chain for Jim's watch.
Third, when Christmas finally comes, they realize that they have gifts that they cannot use, but in the end the narrator breaks into the story and say that they are the wisest. To give is better than to receive and to give in love is to be wise. This is the resolution.