The protagonist in "The Sniper" is an Irish sniper; the antagonist is also an Irish sniper. The title of the story may, ambiguously, refer to either one. As we learn at the end of the story, they are also brothers. Note that within the text of the story, it is not actually explained why they are fighting.
The protagonist sniper is a Republican, and his antagonist brother is a Free Stater. This refers to the political positions held by the opposing factions of the Irish Civil War; the Republicans, like the protagonist, supported complete Irish independence, while the Free Staters supported the terms of a treaty between Ireland and Britain that would have provided for greater degrees of Irish independence, but not complete autonomy (operating completely without connection to the British government).
In historical context, the Civil War split military and political groups that had, until recently, been allied against the common British enemy. This explains why the brothers were fighting in the same place, but on opposite sides, which otherwise would have been a bit of a stretch.