According to Oxford Dictionaries, "madam" is the correct spelling in English. According to Collins French Dictionary, "madame" is the correct spelling in French. Both are used to address a female.
In English, the word "madam" is derived from the Old French phrase "ma dame," meaning "my lady," according to Oxford Dictionaries. Today, the word is typically used as a formal address for a woman, such as in the beginning of a business letter. "Madam" can also be used as a title, such as "Madam President." The word can also mean a woman who owns a brothel, according to Oxford Dictionaries.