A first-person narrative is a story told by a narrator who is writing or speaking directly about themselves. The narrator relates the story by using a lot of I-statements and telling the story as it happened to him or her. A good example of a first-person narrative is Salinger's The Catcher in the Rye. The action of the story is told entirely by Holden Caulfield, who is speaking about his own experiences:
If you really want to hear about it, the first thing you'll probably want to know is where I was born, and what my lousy childhood was like, and how my parents were occupied and all before they had me, and all that David Copperfield kind of crap, but I don't feel like going into it, if you want to know the truth. (J.D. Salinger, The Catcher in the Rye, Chapter 1.)
Less common in literature is the second-person narrative, where the narrator of the story is writing or speaking to You, the second person, as though addressing the reader personally as would be done in a conversation. A good example of this in literature is the brilliant Aura by Carlos Fuentes. In this story, the action is directed at the reader, as follows:
Lees ese anuncia: una ofterta de esa naturaleza no se hace todos los días. Lees y relees el aviso. Parece dirigido a ti, a nadie mas.
You read the advertisement: an offer of this kind doesn’t get made every day. You read and re-read the ad. It seems to be directed at you, and no-one else. (Carlos Fuentes, Aura, Chapter 1.)
Most common is the third-person narrative, sometimes called the omniscient narrator. In literature, the action of stories written in the third person, as well as the internal thoughts and emotions of all the characters, are expressed by the same narrator. As a random example:
The Ladies of Longbourn soon waited on those of Netherfield. The visit was returned in due form. Miss Bennet's pleasing manner grew on the good-will of Mrs. Hurst and Miss Bingley; and though the mother was found to be intolerable, and the younger sisters not worth speaking to, a wish of being better acquainted with them was expressed towards the two eldest. By Jane this attention was received with the greatest pleasure; but Elizabeth still saw superciliousness in their treatment of every body, hardly excepting even her sister, and could not like them... (Jane Austen, Pride and Prejudice, Chapter 6)
Note how in this last example, the narrator describes the external actions of a visit and then the internal impressions and opinions of several of the characters, in quick succession.