“Madam” is a very European term used to address a woman. The French use it quite a bit.

“Ma’am” is more common in most states. It’s used to be polite, especially when one is not sure if the woman in question is married, and how she should be properly addressed. It is often used to address an older lady.

“Miss” is also quite common, and is often used when addressing a younger woman. Again this term is used when one is unsure of the marital status of the woman.

If a woman is married, she is addressed as “Mrs.”

When one is not sure, a woman can be addressed as “Ms”.

These terms are used when writing lettters, especially.

When these terms were in more active usage, any female would use Sir upwards while a Madam or Miss would be downwards terms from a man to a woman as men are always of higher standing. These terms would always be used downwards by anyone with higher status, such as a Lord or a Lady, however.