The story revolves around the spectacular diamond necklace that Mathilde borrows from Mme. Forester for a ball. That set of jewels gives Mathilde the best night of her life. It also ruins it a few hours later, when she loses the necklace and spends the next ten years paying to replace it. And it's also thanks to the necklace that we get the final, gut-wrenching twist at the end.
Interestingly, the original French title, "La Parure," doesn't exactly mean "The Necklace," although that's how it's always been translated. It actually means something more like "the finery," or "the set of jewels."