At the beginning of the story we learn that Mr. Woodifield returns to the city, every Tuesday, to visit his friend, known only as "the boss." It is implied that they are friends by how they behave around one another and speak to one another. For example, Mr. Woodifield sits in a chair opposite the boss, "smoking a cigar," while the boss "roll[s] in his office chair." Both Mr. Woodifield and the boss are evidently very relaxed and comfortable in each other's company. Mr. Woodifield also speaks about the boss's office "wistfully, admiringly." The boss feels pity for Mr. Woodifield and thinks of him as a "poor old chap." He offers Mr. Woodifield a glass of whiskey, remarking that "it's beautiful stuff. It wouldn't hurt a child." Thus the way in which the boss speaks to Mr. Woodifield, and the way in which Mr. Woodifield speaks to him, suggests that the boss is Mr. Woodifield's friend.
Later in the story we find out that the the boss's son died in World War I. He found out about his son's death, six years prior to the events of this story, when a telegram arrived for him. He recalls from the telegram the phrase "deeply regret to inform you." We learn that the boss was devastated by the death of his son and that he feels guilty and ashamed for not feeling the same depth of grief six years later. He thinks that there is something "wrong with him" because he no longer feels "as he want[s] to feel." The implication here is that the father thinks that it is wrong and abnormal of him not to still feel as acutely grief-stricken as he did when he first found out about his son's death. He is tortured by this feeling of guilt. As a character, therefore, the boss is defined mostly as the father of a son killed in the war.