While each hotel defines its family room its own way, it usually refers to rooms available in the hotel that comfortably sleep at least 4 people. A family room is usually a larger room with 2 double/queen beds, or 1 double/queen bed and 2 twin beds. At some hotels, you might find 1 double bed for the parents and a sofa bed or bunk beds for the children.
The family room may also include other family-friendly conveniences such as infant beds, rollaway beds, a ground-floor location for convenience, treats for the kids, microwaves or refrigerators for meal preps.