In the original French, a bourgeois was originally merely a free inhabitant of a bourg, or town. Through a natural evolution it became the label for members of the property-owning class, then of the middle class. As an adjective it is used with contempt by bohemians and Marxists to label conservatives whose views are not sufficiently revolutionary.
The class made up of bourgeois (which is both the singular and the plural form) is the bourgeoisie. Shaky spellers are prone to leave out the E from the middle because “eoi” is not a natural combination in English, but these words have remarkably enough retained their French pronunciation: boorzhwahand boorzhwazee. The feminine form, “bourgeoise,” is rarely encountered in English.