First Person and Third Person (Limited Omniscient)
The first three stories—"The Sisters," "An Encounter," and "Araby"—are told in the first person, but the rest are told in the third person (limited omniscient). Joyce is famous for using a little thing we like to call free indirect discourse, which is a fancy term for a type of third person in which the language that is used to describe a character's thoughts and feelings sounds like the character's very own style of expression. And it's lucky he does use it, because these characters sure don't like to talk about their feelings; they play their cards super close to the vest. So free indirect discourse is one of the only ways we get access into their inner worlds.
Getting Inside Their Heads
This is particularly noticeable in Dubliners in stories like "Eveline" and "A Little Cloud." Even though the narrator isn't Eveline, the language the narrator uses seems appropriate for a girl who is too young to make the decision she faces. For example, when Eveline imagines the future, she thinks about how much respect she will get when she is in the new country: "But in her new home, in a distant unknown country, it would not be like that. Then she would be married—she, Eveline. People would treat her with respect then."
Check it out: this isn't the most literary writing in the world. There aren't any big words (as there are in stories like "The Dead,") and non-specific pronouns like "that" and "then" take the place of the specific details that we know Joyce could have used if he wanted to. It's also not a very high level of thinking: obviously, being in a new country will bring lots of difficulties, too, but Eveline can't really grasp the complexity of the situation all at once. In short, this sounds exactly like we'd expect a young girl to sound—if we could read her mind, that is.
So why bother with all this free indirect discourse in the first place? Probably because Joyce wants to capture the way that his characters themselves think and express themselves—in this case, in short sentences and small words. But we also get the best of both worlds. We see how Eveline thinks, but we don't get stuck in her brain. We get really close the character's psyche, but we stay just outside, which gives the narrator a little freedom to roam.