The narrator in this story is a man named Nick Carraway. He is from a pretty well off family that lives somewhere in the Midwest. He is related to Daisy Buchanan, who is one of the major characters in the story.
He is able to function as narrator because he ends up living right next door to Gatsby. Because of that, he meets Gatsby and can narrate about him. Because he is related to Daisy, he meets her husband and Jordan Baker and the Wilsons and can talk about them.
He functions as a first person narrator, which gives us insight into his attitudes and feelings about the other characters.
Just to add to the above answer, you can find evidence to the narrator's last name in the fifth paragraph of the novel.