'Madam' is a formal term of address in English. It is the feminine version of 'sir'. Airline staff and hotel staff, for example, would say 'Good morning, madam' to greet an adult female guest.
'Madame' is the French equivalent, but it is also the equivalent of the English 'Mrs' in that it is used as part of a name. The MD of the IMF, Christine Lagarde, for example, would be referred to as 'Madame Lagarde'.