Maupassant was a student of the great French author Flaubert, who was a founding figure of "Realism" (with a capital "R") as a literary genre. Realism meant more than just writing about real-seeming situations in a realistic way—it often meant writing about "average" people—not super-rich, or famous, or holy, or good, or even happy people. Not terribly exciting people either.
Usually the "average" person meant a middle-class person, and particularly the bored, unhappy middle-class person who longs to live the more exciting life of the rich and famous. Sound familiar? You might think of Flaubert's classic Realist character Madame Bovary as an inspiration for Mathilde.
It also seems fair to call "The Necklace" literary fiction. Maupassant was a big-time innovator of the short story as a genre of literature. He's known particularly for his unique talent for creating compact plots, and this story is one of his finest productions. Not only that, he practically invented the twist ending—and there's almost no twist ending more famous than his one.
Finally, if you think "The Necklace" has a clear moral message (for example, "Be honest," or "Wealth is always false"), you might want to call the story a parable, which is a simple work meant to illustrate a "moral." (Then again, you might not think the story has an obvious moral. You might even say that the uncertainty about whether it does means that it's not a parable, because the moral of a parable is supposed to be obvious. We'll leave that one up to you.)