There are several themes, or key points about life (and the world) in general, that the author makes in the short story. It is important to look at themes within the historical context of the story, so keep in mind the specific setting. This story takes place in Louisiana, after the Revolutionary War, but before the Civil War. Slavery is very much legal and still largely supported and practiced in the United States.
Prejudice against slaves was rampant during this time. White slave owners, and many others, not only believed that black slaves were not equal, but it would be safe to say that many even considered them less than human. Because of this, slaves were treated as property, not people.
One key theme of this story is this sense of human prejudice and superiority of one human over another. The author structures the plot around the mystery of identity. Neither the audience nor the characters know for sure whether the mixed ancestry of the baby comes from his mother or his father. Of course, Armand assumes it comes from his wife, which she accepts as a blow of hatred and scorn.
A quick conception of all that this accusation meant for her nerved her with unwonted courage to deny it.
By establishing this mystery of identity from the very start, Chopin is able to show raw prejudice and complete rejection. Armand turns his back on Desiree and his son, based on racism. In this way, the author reveals a very dark side of humanity and America's history. She then takes this message even further with the ironic plot twist at the end of the story, which is that the one most cruel and prejudiced character throughout the story, Armand, is actually the one who has held the mixed ancestry all along. He possesses it, physically inside him, and is still completely blinded by his sense of prejudice and superiority, which causes the demise of his family. Armand's fate is left to the interpretation (and judgement) of the reader.
...that our dear Armand will never know that his mother, who adores him, belongs to the race that is cursed with the brand of slavery.