The term is often used to a non-monarchical heads of state or chief executives who don't have that kind of style in their own country. Some countries have a title, official or unofficial, that is or can be translated as first lady. The title is not normally used for the wife of a head of government who is not also head of state.
First Gentleman is the male equivalent of the title in countries where the head of state's spouse has been a man, such as the Philippines or Malta. While there has never been a male spouse of a U.S. president, "First Gentleman" is used in the United States for the husband of a governor.
First Spouse, a rare version of the title, can be used in either case where the spouse of a head of state is male or female. This term is used to promote gender equality and gender neutrality.
In the United States, collectively, the president of the United States and his spouse are known as the first couple and, if they have children, they are usually referred to as the first family.