ofdirect characterization, O. Henry establishes the degree to which both partners treasure their hair and watch respectively. In this manner, they are established as their most treasured possessions:

Now, there were two possessions of the James Dillingham Youngs in which they both took a mighty pride. One was Jim's gold watch that had been his father's and his grandfather's. The other was Della's hair.

Note, however, that regardless of the degree to which Della treasures her hair, this does not prevent her from selling her hair to buy a gift for her husband, and the same applies to Jim, who has acted similarly in selling his watch. In short, each has sacrificed their most treasured possession for the other, placing the other's needs and desires above their own. In this manner, through their own actions regarding their most precious possessions, they illustrate their mutual selflessness to one another.

In one example of indirect characterization, the narrator tells us

. . . whenever Mr. James Dillingham Young came home and reached his flat above he was called "Jim" and greatly hugged by Mrs. James Dillingham Young, already introduced to you as Della. Which is all very good.

We learn, indirectly through their behavior toward each other, that Jim and Della are a very loving couple. They care a great deal for each other, and, despite their financial difficulties and challenges, they each seem to feel so fortunate and rich in love. The narrator does not directly tell us this but, rather, suggests it indirectly via the characters' actions. We learn that Della is a bit vain in a similar manner. After she finishes crying, early on, she "attended to her cheeks with the powder rag," and we also know her hair is her most prized possession. The narrator does not tell us she is vain, and it isn't a terribly huge fault, but Della obviously cares about her appearance (though not as much as she loves her husband). She cares whether Jim "think[s] [she is] still pretty."

Later, the narrator describes her as trying to fix up her now very short hair to repair "the ravages made by generosity added to love." Now, Della is directly characterized as generous and loving: she was willing to part with her most prized possession—and affect her daily appearance despite her slight vanity —in order to do something nice for her husband.

A good example of direct characterization in "The Gift of the Magi" would be Henry's detailed description of Della's beautiful hair, how it ripples and shades like a cascade of brown water, reaching below the knee almost like a garment. This is important information because it's pertinent to what happens in the story. For Della sells some of her gorgeous locks to a fancy salon so that she can buy Jim a gold watch-chain for Christmas. Direct characterization is used here to show us precisely what it is that Della's prepared to sacrifice to buy a special gift for her husband.

As regards indirect characterization, the description of Jim and Della's shabby apartment tells us a lot about how poor they are. The letter-box doesn't work, the doorbell doesn't ring; their couch is described as "shabby." Also, we're informed that Jim takes home $20 a week instead of $30 as was previously the case. Even in those days that wasn't much to live on.

It can be difficult at times to distinguish between indirect and direct characterisation. However, direct characterisation can only ever occur in a story with a third person point of view, as it is when the writer tells us directly about what a character is like or what a person's motives are. Indirect characterisation is anything that allows us to infer something about a character, but it is not stated directly.

For example, in the story, we can tell through the setting and action that Jim and Della are poor. Consider how their apartment is described:

A furnished flat at $8 per week. It did not exactly beggar description, but it certainly had that word on the lookout for the mendicancy squad.

This is indirect characterisation - the author does not tell us directly that they are poor, but it is made clear indirectly through their setting.

However, later on, the author uses direct characterisation to tell us about the motives of Della and her purpose for saving money:

Tomorrow would be Christmas Day and she had 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.

Note here how the author directly tells us her motives for saving the money and how much she loves him. This is clearly an example of direct characterisation, as it directly reveals to us information about Della and her motives for trying to save money.

Hope this helps! Good luck finding more examples in this great short story.

In addition to the physical characterization mentioned in the previous response, this story also includes a few moments of direct characterization to describe what a character is like internally. O. Henry tells us that as soon as Della sees the chain, she knows "that it must be Jim's" because it resembles him. Henry writes, "Quietness and value--the description applied to both." Henry directly tells us here what Jim is like--quiet, and of value to Della.

Another example of direct characterization in this story is of the Madame, who cuts off and buys Della's hair. The Madame is "large, too white, chilly." This direct characterization is especially important because it is in contrast to Della's direct characterization just a bit earlier in the story--Della is described as "slender," as opposed to the Madame's largeness. Henry also tells us that Della has a "brilliant" sparkle in her eyes, which is much different than the cold, pale characterization of the Madame.

Direct characterization is where an author, or narrator, tells readers exactly what a character is like (both internally (emotionally/mentally) and externally (physically)). In indirect characterizations, readers must infer (make an educated guess) regarding a character's character. Hints are given to the reader through private thoughts, dialogue, and actions. 

An example of a direct characterization provided in O. Henry's "The Gift of the Magi" appears in the section where the narrator is describing Della's hair: "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." Here, the reader does not need to infer about anything. Della's hair is described in great detail. It (her hair) is wavy, shining, brown, and extremely long. Later, after she cuts her hair off, it is described again (after she styles it): "with tiny, close-lying curls that made her look wonderfully like a truant schoolboy." Once again, readers do not need to infer about what Della's hair looks like; the narrator is very specific regarding the details.