Most people will make a distinction between a living room and a family room. The living room is nearly always on the ground floor, is fairly large, and was originally intended as the place where the family sat to read, listen to the radio (in the dear old days of yore), watch TV (later) and entertain visitors if there was no parlor (see other thread). The family room is a less formal room, now nearly always containing the TV, and has, for the most part, replaced the den, which was where the TV usually was between the era of the living room and the family room. The family room may be anywhere in the house, including upstairs or in the basement; it is generally furnished in informal fashion whereas the living room has more formal furniture, rugs and carpet. Most new houses being built today, unless they are large and commodious, have family rooms and no living room or dining room (modern families don't dine in formal fashion; meals are eaten in a section of the kitchen, if not out of hand or lap in the family room). A rec room, or recreation room, is also somewhat different from a family room, as it generally contains some kind of recreational equipment, such as a pool table or pingpong table. (I almost wrote "billiards table", but realize that you would have to go far and wide to find a real billiards table, as hardly anyone plays billiards any more, as distict from pool.)
Also, to add to the list of rooms, there is the powder room, which is a small room with a toilet and washstand, or vanity, but no bathing facilities, usually downstairs so as to obviate a trip upstairs to use the bathroom.
Also, there is, mostly in older houses, the vestibule, which is not really a room, just an enclosed area behind the front door before you get to the hall, which serves to keep out drafts and provide someplace to shake off your umbrella or raincoat and wipe your feet before you mess up the hall.
