Both
madam and
madame are used to describe women, but not as interchangeably as some may think.
Madame is primarily used as the French equivalent of "Mrs.", denoting a woman's marital status. It can also be used to denote respect to an older woman of distinction, who is of a heritage other than American or British.
"It's a pleasure to finally meet you, Madame Giry. Christine has spoken of you kindly."
Madam, on the other hand, is the word that will be used more often.It can be used to convey a woman's rank or authority.