Joyce was a modernist writer, meaning he was consciously breaking with traditional ways of writing; he was experimenting with new forms. Modernist writers were becoming more interested in capturing the interiority of their characters, and this novel reflects that urge. Though told in the third person (with a bit of first-person near the end, in Stephen's diary entries), it is entirely from the point of view of the main character Stephen Dedalus and records what he witnesses using a stream-of-consciousness technique. It is as if a video camera is in Stephen's head, filming everything as he experiences it in a raw, unmediated way. It is the story of Stephen growing up and maturing into an artist, but unlike, say, Dickens in a novel such as Great Expectations, which is also about the maturation of a boy into a man, Joyce doesn't pull the camera back, so to speak, and doesn't give us a wider context. Dickens will show a scene as experienced by the young Pip, then include Pip ruminating on it with his adult consciousness. Joyce simply lets the reader see what Stephen sees at whatever age and does not try to interpret these experiences for us. This brings us closer to Stephen but also leaves it more up to us as readers to sort out and interpret what is happening.
James Joyce is well known for his unique storytelling abilities, and this novel reinforces this.
Joyce uses a third-person point of view, but a very unique one. Events are not told in chronological order, for example. Also, the narrative focuses on its protagonist, Stephen Dedalus. What is interesting is that Joyce's diction is directly related to Stephen's age. For example, at the beginning of the novel, he is very young, perhaps a toddler, so Joyce's diction reflects that.
Also, another unique thing about Joyce's narrative is that "[h]is narrative is narrow and tightly focused; he does not tell what is happening but rather tries to show what is happening without explaining the events that he is showing" (Enotes).
Finally, Joyce also uses stream of consciousness (writing as one thinks, which can be very fragmented) and interior monologue (a procession of thoughts in one's mind when one thinks to themselves).