In modern French, le fauteuil is simply an armchair; in English, a fauteuil is a specific type of armchair, although it is still somewhat broad in meaning. A fauteuil is an armchair that has a wooden frame, usually clearly visible, but with a padded back, seat, and arms. The arms may be rather minimally padded, and both the arms and the framing around the upholstered back and seat may be as elaborately carved as a picture frame. This style became popular in the late 1600s and early 1700s, and was commonly seen in the French court; as such, many chairs referred to as fauteuil by English speakers are quite stately and elaborate. The arms are usually open, although this is specific to the English usage; in France one can purchase a fauteuil à oreilles ('wing chair') without causing confusion.
It is becoming popular in English to call any fancy armchair with open arms a fauteuil, and any with closed sides a bergère. This is nonsense, but if it makes communication with your decorator easier, so be it.
Fauteuil is also used in French, and very occasionally in English, to refer to a metaphorical chair; to 'occupy the chair' (occuper le fauteuil) is to take the lead in a meeting; the position of the French president is sometimes referred to as the fauteuil. Most famously, it is used to refer to one who has gained membership in the Académie française, with this position often referred to as the fauteuil académique.
In English it is pronounced badly, so don't worry too much. It should sound something like foh'-tay (/foʊˈteɪ/), but this will depend on what part of the English-speaking world you are in and how much you lean into the French accent.