La Blanche is mentioned only in passing in this short story and only three times, but she is an important character, as she lends considerable context to the conflict of the story.
La Blanche's name is interesting: it means "the white one," even though she is clearly a Black slave who is the mother of a "quadroon boy." La Blanche's name implies that she is of mixed race and possibly that she has pale skin. If this is the case, she is considered to be Black only because she has Black heritage, not because of her skin color: something which is very important contextually.
When Désirée implores Armand to recognize that she is a white person with white skin, he says mockingly that her skin is indeed "as white as La Blanche's." The implication here is that it does not matter whether Désirée appears to be white: if she has an element of Black blood in her, then she and her child, like La Blanche, are deemed to be Black and seen as inferior.
It is also possible to argue that La Blanche is Armand's mistress. Désirée does not seem to notice anything odd about the fact that Armand has been visiting La Blanche's cabin, but to the reader, the detail is strange. There is a connection indicated, too, between La Blanche's child and Désirée's, which might be because both appear to be Black—or because both are the children of Armand.