This is an interesting story in terms of its structure and emphasis. Like much of Bierce's work, it seeks to expose the true tragedy of civil war, which in this case is portrayed through the story of the Grayrock brothers.
The introduction to the story, the first element, is easily enough identified. In this section of a story, the writer provides the setting—which Bierce does explicitly, giving the place (southwestern Virginia), the year (1861), and even the time of day (a Sunday afternoon). He then introduces the protagonist, Private Grayrock of the Union army, who is posted "as a sentinel" in the forest. We now know the context of the story and the main character with whom we are concerned.
The next part of the plot is the rising action. Again, in this story, we can identify this part quite easily—the rising action is where we see some indication of what difficulty the character will face and what will cause it. We learn quickly that the "gloom of the wood" is deep on this night and shortly discover that Private Grayrock is lost. This begins to present a challenge for him as he meanders around the unfamiliar undergrowth. The tension heightens until such time as Grayrock encounters an unnamed figure in the undergrowth—and fires.
At this point, we enter the climax of the story. Commencing with the shot itself, the climax continues into the section in which we see Grayrock battle with himself mentally: he is particularly concerned that he cannot find the body of the man he is sure he has hit, and we see some indication of what mental, as well as physical, challenge he is facing here. Grayrock does not know if the man he has hit is friend or foe, and this troubles him. When the general allows him to leave camp the next day, he is conflicted, because this leave is granted on account of his bravery, and meanwhile Grayrock is guilt-ridden and anxious because he cannot find the person he has killed, and there is something in his subconscious which cannot overcome his concern about this.
The falling action in this story comes in the shape of a dream which betrays Grayrock's subconscious concerns and gives some clue as to what the outcome of the story will be. The foreshadowing in this dream is significant. Having wished aloud that he could "find my man," he dreams of his brother. We learn that Grayrock had a twin who was adopted by an "enem[y]" of the man who adopted Grayrock, although the two were kinsmen. This outlines one of the key conflicts and concerns in the story: the idea that civil war pits brother against brother. The "song of the mocking-bird" runs through Grayrock's dream as he imagines his brother gone forever, and the reader is uncomfortably sure now of what the denouement will be.
Sure enough, we now come to the conclusion or denouement of the plot, where the reader discovers the outcome. Moving away from his resting spot, William Grayrock, our protagonist, comes upon the body he has sought at last and finds that "his man" is, of course, his brother. This tableau is painted as a "masterwork of civil war," and as the "shrilling bird" sings, we learn that William Grayrock did not return to the army for roll-call that night, "nor ever again there-after." The suggestion is either that he becomes a deserter, unable to continue fighting such a war, or that he kills himself.