There are several theme subjects presented in this short story. I'll give you some ideas to get you started:
1. Appearance vs. Reality: consider Madam Loisel's lack of contentment. She is described as beautiful, and we learn later how comited her husband is (to stand by her side and work with her through a stupid and terrible mistake), yet she constantly wants what she does not have. She appears (to some) to have two very covetous qualities - but she is not content. Then of course, the necklace itself is a symbol of this theme.
2. Class distinction: Mm. Loisel's primary problem is her jealousy of those who have more than she does. This presents the theme of social class differences and the conflicts that arise as a result.
3. Getting what you deserve: I didn't word that well, but you get the idea. This theme is obvious by the question presented at the end of the novel - does Mm. Loisel get exactly what she deserves? Does the reader pity her in the end?
Consider also some common threads throughout the three bigger subjects presented: greed, jealousy, discontentment, gratitude (or lack there of). Hopefully this gives you something to chew on and then write about.
One very pervasive them in the story The Necklace is the consistent preocupation with appeareances and the idealism about what happiness is.
In Mme Loiselle's case, she is consistently complainting about not having what she deserves. To her, however, what she deserves is simply showing off a big house, servants, a lot of money, and success.
She is taking for granted the things she DOES have such as health, a home, food on the table, and a husband that cares for her.
Therefore, the necklace itself (a fake copy of an expensive diamond necklace), is something that, to her, prepresents all that she wishes for: A glitzy and shiny attractive lifestyle - albeit, one that gives her a fake sense of happiness.
The previous post touches upon what is common trope for Guy de Mauppassant: the pettiness of the bourgeoisie and his naturalistic "vague and unconvincing" relationships of men and women effected from his observation of his fellow Normans and his fellow workers as a civil servant, both of whom he found petty.
In the marriage of M. and Mme. Loisel there is little true affections and virtually no interaction. M. Loisel clearly idealizes his pretty wife and tries to make her happy whether it is by procuring tickets to a ball or by sacrificing his savings for a rifle so that she may be able to purchase a new dress for the ball. Her insensitive reactions point to the emptiness in the marriage of the two. Continuing in this vein, Mme. Loisel does not bother to even thank her husband for the great personal sacrifices that he makes over so many years.