In one or two sentences: A woman's over-preocupation with social status disconnects her partially from her much humbler reality. Her own irrational obsession will lead her through a careless mistake that will ultimately ruin her life.
In my opinion, the conflict in this story is of the type where the character is in conflict with her situation -- her place in the world. Mathilde Loisel's ambition is in conflict with her circumstances.
The conflict is between her circumstances and her aspirations. She is relatively poor (though she has enough money to have a maid and to go the theater) but she wants to be a member of high society.
It is this conflict that causes the story to happen. She wants to appear rich and important so she borrows the necklace. Then she destroys her life trying to replace it when she loses it. So her downfall is caused by this conflict.
The conflict in the story "The Necklace" centers around greed and vanity. The main character's greed and vanity leads to her demise, and to lead a miserable life of hardship and woe. If she had been content with the blessings that she had in the first place, she would have never had to spend her entire life paying for the sacrifice that her greed demanded.
The main conflict, in literal terms, deals with the fact that Madame Loisel loses a very expensive necklace that a good friend loaned her, and has to spend her entire life working hard to earn the money to buy a replacement. The working drives her into poverty and ruins her life, at the same time teaching her valuable lessons about the value of hard work and earning one's own bread. It is a rather sad tale, especially considering the ending, when her friend, Madame Forestier, informs her that the lost necklace was worth next to nothing.
I hope that those thoughts helped good luck!
Madame Loisel suffers from internal conflict, she is angry that she has allowed herself to be "poor".