What is the external conflict in "The Gift Of The Magi" by O. Henry?

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.

"The Gift of the Magi" is a wonderful story about unconditional love. The external conflict facing Jim and Della in the story is their poverty, their need for money. It is Christmastime, and each wants to give a special gift to the other. However, they are very poor and don't have the money to buy the gifts that they want to give. So in order to earn the money they need, each makes a sacrifice, only to discover that they can't use the gifts because of the sacrifices they each made.

In "The Gift of the Magi," what are the external and internal conflicts?

Clearly conflict is one of the major ingredients in this excellent short story, which is normally used to teach irony in schools. The major source of external conflict that Della faces is against poverty:

Tomorrow would be Christmas Day and she had only $1.87 with which to buy Jim a present. She had been saving every penny she could for months, with this result... Only $1.87 to buy a present for Jim. Her Jim. Many a happy hour she had spent planning for something nice for him. Something fine and rare and sterling - something just a little bit near to being worthy of the honour of being owned by Jim.

Della thus faces a conflict of how she is going to buy the present she wants to buy for her husband, whom she loves so much.

The internal conflict that Della faces is when she wants to sell her hair to gain the money to be able to buy the present for Jim. Note how the text describes this internal battle:

So now Della's beautiful hair fell about her rippling and shining like a cascade of brown waters. It reached below her knee and made itself almost a garment for her. And then she did it up again nervously and quickly. Once she faltered for a minute and stood still while a tear of two splashed on the worn red carpet.

The tears of course are evidence of the cost of this conflict, as she chooses to sacrifice her one possession in which she took "mighty pride" for the sake of her husband, so that she can buy him the present that she wants for Christmas.

What type of conflict does Della face in O. Henry's "The Gift of the Magi"?

I would say that Della's major conflict is actually of the character vs. character variety. She worries a great deal about what her husband, Jim, will think of her once all her beautiful hair is gone, but she only "faltered for a minute" and lets a few tears fall when she fights with herself about whether or not to actually sell her hair. It does not take her long to make a decision, and, once she does, she acts immediately. However, Della says to herself regarding her husband:

If Jim doesn't kill me [...] before he takes a second look at me, he'll say I look like a Coney Island chorus girl. But what could I do?--oh! what could I do with a dollar and eighty-seven cents?

When Jim does walk in, he has an expression on his face "that she could not read, and it terrified her." She fears that he will not think she is pretty anymore now that her hair is gone. His response, looking around the room with a rather vacant expression and "an air of almost idiocy," makes her really nervous that he is angry with her. This conflict is more imagined than it is real, of course. Jim is only shocked—especially because of his gift to Della—and not angry. However, we do not know that right away, and neither does she.

Della desperately wants to buy Jim a nice gift for Christmas; she wants to buy him a nice chain to go with his pocket-watch. But, unfortunately, she doesn't have anywhere near enough money to pay for it. Della and Jim are dirt-poor and neither can afford to buy the gifts that they really want. Della deals with this problem by selling some of her beautiful locks of hair to a fancy wig-maker. This raises enough money to pay for Jim's watch-chain. Unbeknownst to her, however, Jim has decided to deal with his own conflict in a similar fashion. He's sold his pocket-watch to buy Della a gift of some fancy combs. So both Jim and Della, in dealing with their conflicts, have ended up buying each other what turn out to be useless gifts.