Explain the role of Madam Valmonde in the story.

Madame Valmonde is Desirée's adoptive mother. She could not bear children of her own, so she and her husband adopted Desirée after finding her abandoned, at a toddler age, and asleep "in the shadow of a big pillar."

The prevailing belief was that she had been purposely left by a party of Texans, whose canvas-covered wagon, late in the day, had crossed the ferry that Coton Mais kept, just below the plantation.

As such, her role as mother and nurturer of Desirée is established. She also had the added role of serving as the person responsible for safeguarding the best interests of Desirée in her Bayou society. Hence, not knowing Desirée's true origins was problematic in the sense that there was always the possibility that she was of mixed racial backgrounds, which is a cause for social inequality. It also carries with it the potential of her being socially shun and never able to find a rich husband. Since Desirée appeared to look like any other mainstream high society white woman, she was able to marry well.
Hence, when Madame Valmonde goes to visit her daughter, she takes a good look at Desirée's child to verify if he inherited any African traits. When Valmonde sees that the child does seem to be mixed, she first suspects that the heritage comes from Desirée's line. For this reason, when Armand shuns Desirée and kicks her out of his life with his child, Madame Valmonde responds like a good mother, telling her daughter to come back to the family plantation immediately with the child. 

"My own Desiree: Come home to Valmonde; back to your mother who loves you. Come with your child."

We know that Desirée kills herself and her baby. As a result, Armand makes a huge bonfire burning all of his wife's belongings, letters, and also a letter from his mother to her husband, where she confesses of being of the race that was "cursed with slavery."