The titular sniper, who is not given a name, is a young Irishman. We are told little about him directly; most of his characterization comes from the description of his actions, and what he looks like, not necessarily who he is, or was, before the war. This is probably intended to diminish his individuality, with the purpose of making him a representative of virtually any young man in the war.
The first description we are given of the sniper is that he has the "face of a student, thin and ascetic". While allusions to a student may draw up the idea of youth and innocence, the use of the descriptors "thin and ascetic" (ascetic meaning self-disciplined) implies that a better term might be scholarly or spartan. Meanwhile, his eyes have the "cold gleam of a fanatic". This implies that the sniper is at war without and within; his self-control tempers a fierce personal motivation.
We can surmise that this is not the sniper's first battle; his careful evaluations of the risks of each action imply that he has done this before, and perhaps seen the consequences that befell others less fortunate than himself. This is also suggested by his ability to care for himself after he is shot, and to overcome the pain of his injury by the force of his own will.
For much of the story, the sniper functions as an embodiment of self-control and restraint, but this veneer is cast aside when the tension has abated, and the sniper is suddenly overcome with disgust. He is, after all, only human, and can only stave off his own emotions for so long under this kind of pressure.
Overall the sniper is meant to be someone we can relate to; who behaves as we might wish we could behave under the circumstances, but who also has realistic weaknesses and makes mistakes. This reinforces the purpose of leaving the sniper unnamed; the reader can put themselves in the sniper's place, and see themselves behaving in the same way.