The conflict in this story is between the boys and society (or you might call it "the way the world works"). The boys want excitement and adventure, but they find that things that seem like they would be exciting either A) become boring routine or B) are strange and frightening.
This is shown most clearly in the part of the story where Mahony and the narrator skip school. They think that it will be exciting, but it becomes boring. Then, things seem to liven up when the old man comes on the scene. But he is weird and somewhat scary and perverted.
The conflict in this story is between the boys' desire for adventure and society (or "the world's") penchant for creating a situation where things are either boring or frightening -- nothing in between.
In this story, there is an evident clash between innocence and experience. The very title refers to a seemingly ordinary encounter between a boy and an old man, but we can also view this story in two more ways. Firstly, it is a story in which the main protagonist has an encounter with the unknown world he wishes to be a part of, of all its challenges and perils it brings and its overwhelming impact on the innocent world of a child. Secondly, the story implies that a boy eventually has an encounter with his true self, realizing that this experience has taught him a lot about himself and the world around him.
The central episode of the story takes place in a field where the narrator and his friend, Mahony, rest after crossing the river Liffey. The episode is focused on the encounter between the narrator and a predatory old man who approaches the boys and reveals some rather alarming thoughts about touching girls, whipping boys, etc. This encounter represents the boys’ first encounter with the adult world and all its mysteries:
At times he spoke as if he were simply alluding to some fact that everybody knew, and at times he lowered his voice and spoke mysteriously as if he were telling us something secret which he did not wish others to overhear.
The protagonist eventually sees the new world as a perilous, threatening and incomprehensible one, which has the power to crush the comfort and safety of the world he is accustomed to. Therefore, his only choice is to return to “a well-established boring old order” which he gladly embraces.