Madame Loisel drives the plot in two very important ways. Firstly, her dissatisfaction at her humdrum, middle-class life leads her to borrow what turns out to be a fake necklace. Mathilde believes that she's entitled to better things and that what she thinks is an expensive necklace is just the thing to express her exalted self-image. For Mathilde, it doesn't matter what you are; it's what people think you are that counts. She's so shallow, so incredibly vain and superficial, that she's convinced that wearing expensive jewelry will be enough to make people think highly of her.
A second way in which Mathilde's character drives the plot is through her lack of real class. Despite what Mathilde might think about herself, she isn't really the displaced aristocrat she'd have us believe. For all her airs and graces, Mathilde doesn't know the difference between fake jewelry and the real thing. This means that she and her husband end up getting into serious debt trying to buy a replacement for a lost necklace that's actually worthless. As a result, the Loisels are plunged into a life of poverty. Now they occupy a position far lower down the social scale than they previously enjoyed. Not only that, but they're further away than ever from the social elite to which Mathilde has always felt she instinctively belongs.