A marquise (pronounced 'mar-keez') chair was originally an extra-wide armchair, barely wide enough to seat two people; it appeared during the rococo period of the mid-1700s, coved and very low, so that the arms and back formed a single U shape with a smooth flow from the arms into the back. While these chairs were well-upholstered and built for comfort, they were not chairs built for relaxing, as you did not have upper back and head support.
While marquise chairs are still sold, modern versions are usually not so wide, and the term refers primarily to the low U shape. Unfortunately, many chairs that are not in this form are also sold as marquise, to the point that the term does not seem to mean much of anything; this is particularly the case with 'marquise side chairs' and 'marquise dining chairs', which often have neither arms nor low backs nor particularly wide seats.