In James Joyce's writing an epiphany is a sudden revelation of truth about the characters' inner life or their place in society. These moments are central to the development of the stories and, as the essay by Francesca Valente quoted below (second link) argues,
the central meaning of Joyce's works is provided not so much by plot but by the revelation they suggest of a certain universe in a certain order.
So it is crucial that you identify these epiphanies to gain a full understanding of the stories. This is the result of the modernist focus on character psychology and the study of the human mind/consciousness rather than on a linear and sequential plot development. The three links that I have included below take you to in-depth analyses of each story in Joyce's Dubliners including their epiphanies.