The first conflict is within the sniper himself. He is young and therefore should be at least somewhat innocent and inexperienced. But he finds himself in a civil war, faced with gruesome experiences of violence and death.
His face was the face of a student, thin and ascetic, but his eyes had the cold gleam of the fanatic. They were deep and thoughtful, the eyes of a man who is used to looking at death.
Young, perhaps once full of hope and open minded, he is also a single-minded soldier in the Irish Civil War. His conflict also lies in the fact that he is fighting his own countrymen - "brother against brother" as civil wars are often called. This resurfaces at the end of the story.
The general conflict in the plot is the war itself. Defending himself, the sniper feels he must shoot an old woman who has identified his position. He doesn't have time to react and/or reflect on killing the woman because the enemy sniper begins firing at him.
However, after killing the enemy sniper, his internal conflict emerges again. "The sniper looked at his enemy falling and he shuddered. The lust of battle died in him. He became bitten by remorse." Frustrated, he throws his gun and it fires, almost hitting him. This jars him out of remorse back to his focused state as a soldier and he even laughs, perhaps out of nervousness or simply because his emotions are all over the place and he doesn't know how to react. In the end, the general conflict of civil war and his internal conflict of duty/remorse resurface when he looks at his enemy and sees his "brother." This can be taken literally or figuratively since civil wars are often considered wars of brother against brother: wars fought between fellow countrymen.