Bergère is French for shepherdess, and it became a fashionable word in the mid-1700s to indicate that something was part of the rustic aesthetic. A 'bergère' might be a bergère hat or a bergère gown (or Robe en bergère), but most often the term, when used on its own, will refer to a bergère chair.
To be clear, while some things that were called bergère might indeed be somewhat related to peasant fashions, what was really being signaled was that the object was modern and fancy enough for Marie Antoinette or other French trend-setters to consider fashionable. The bergère chair is a rather extreme abuse of the word, as it is one of the least likely seats for a sheep-minder to use.
A bergère is an upholstered French armchair (a type of fauteuil) with a visible wooden frame and upholstered arms, back, and seat, with the seat being over-upholstered or having an unattached cushion. While it is likely to be quite fancy, with carved arms and legs and fine upholstery, it is also intended to be quite comfortable and fairly easy to move around... at least, by the standards of Georgian era mansions.
These chairs came in a number of forms, with bergère à oreilles having high sides that could protect the occupant from drafts; bergère à la reine with a more traditional straight back; bergère en cabriolet, with the back curving to flow smoothly into the arms; or a bergère marquise, now commonly called a marquise chair, a wide seat with a low back that is often not much higher than the armrests. However, as bergère chairs have been continuously produced for 300 years, at this point the term can be fairly fluid in its usage; in modern times any stiff, fancy upholstered chair with at least some part of its wooden frame (usually legs and part of the arms) showing might be referred to as a bergère. Many modern sources will claim that a bergère chair will have closed and fully upholstered arms, while a chair with open arms should be referred to as fauteuil; I will leave modern English digest this for a another hundred years before I accepted it as common usage.
In English, it is common to see both bergère and bergére, as English speakers are not attentive to such details. In this case, the grave accent is correct.