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.
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.