The conflict is that Mathilde loses the necklace and has to give up the one thing she has to give it back.
A conflict is, simply put, a struggle between opposing forces. Conflicts can internal or external. An internal conflict is between a character and his or her self, and usually involves a tough decision or fear. An external conflict is between a character and an outside force, such as another character, society, nature, or technology or the supernatural.
In “The Necklace” Mathilde does not have much money, but she has great beauty. She always wishes for her money to match her beauty, but she does not marry high enough in society to accomplish this. One day she borrows a necklace from a wealthier friend to attend a society ball, and promptly loses it.
Thus enters the main conflict of the story. Mathilde must decide if she should tell her friend she lost of the necklace, of course, but it never occurs to her. Her pride forbids it. So she and her husband borrow money to replace the expensive necklace, and work it off.
Thereafter Madame Loisel knew the horrible existence of the needy. She bore her part, however, with sudden heroism. That dreadful debt must be paid. She would pay it.
For ten years, they continued this way, and Mathilde grew older and less beautiful. She struggled with the consequences of poverty, but it was an internal struggle. It was a struggle of guilt and shame, not because of losing the necklace, but because she never had the money to replace it.
In the end, Mathilde’s struggle turns out to be an ironic one. The necklace she replaced the lost one with was real, but the lost one was fake. Madame Loisel, her friend had no idea of what Mathilde went through. If Mathilde had not had such pride, she would not have had to struggle at all. In struggling, she lost everything she had to be proud of. Such is the irony.