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.