This is a great question. There is only one main conflict in the story and this conflict is beautiful and as O. Henry states, filled with wisdom. In order to understand this conflict, a little context is necessary.
Della and Jim, a young married couple, are the protagonists of the short story. Moreover, they are poor. However, they want to get each other a meaningful gift at Christmas. Della has one prized possession - her long and beautiful hair. Jim also has one prized possession - his pocket watch. So, what do they do? Jim sells his most prized possession to get a set of combs for Della beautiful hair. And Della cuts and sells her hair to get Jim a nice chain for this pocket watch. In the end, both Jim and Della have gifts that they cannot use.
However, the story is beautiful, because there is one conflict that is central. Both Jim and Della want to outdo the other in love. It is a conflict of loving more. In the end, both stand to gain. This is why O. Henry ends his story with the point that they are wise; they are magi.
Here are the closing words of the short story:
"The magi, as you know, were wise men--wonderfully wise men--who brought gifts to the Babe in the manger. They invented the art of giving Christmas presents. Being wise, their gifts were no doubt wise ones, possibly bearing the privilege of exchange in case of duplication. And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days let it be said that of all who give gifts these two were the wisest. O all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the magi."