A fiancée is a woman to whom a person is engaged to be married. A male to whom a woman is engaged is called a fiancé. The "e" is generally added to the feminine form in French, as is the case for fiancée. The pronunciation of fiancée and fiancé is the same.
The terms fiancé and fianceé are borrowed from French and derive from the French verb fiancer, which means "to get engaged to be married". The origin of fiancer is from Old French and originally meant "to promise".
The correct pronunciation of fiancée is fee-ahn-seh and not fee-ahn-say, as is often heard by English speakers. In the audio pronunciation, you will hear that the -ahn is pronounced with a nasal "n", however, it is not absolutely necessary to pronounce it this way to achieve a good standard of pronunciation. What is most important is to pronounce fiancée (or fiancé) with an open ending, with a -seh sound, rather than closing it with -say.