A bergère chair is a French-style arm chair with a large upholstered and/or cushioned seat, an exposed wooden frame, and closed sides. The back and arms are usually upholstered, but they can also have cane work. The back and arms are always closed, rather than open like a fauteuil chair.
Even though they can look a bit formal to 21st-century eyes, bergères were designed to be comfy lounge chairs during the time of Louis XV and Marie Antoinette. That's why they often have matching ottomans — for maximum lounging. The one pictured above is from the Palace at Versailles where the French royal family lived until the French Revolution.
Traditionally, a bergère chair is pretty girlie. Delicate, usually covered in a fussy fabric, comfy but petite. Not a manly chair, and that's fine. But sometimes a more manly space needs that combination of comfy and formal that a bergère chair can bring.
Bergère chairs aren't just for French country girlie rooms any more.
There are very modern bergères (Decorati Access showed a nice modern bergere in a design showcase last summer) and you see them in eclectic and cottage-style spaces, as well (there are some nice examples in the book Flawless Interior Decorating on Google Books).
Or you can choose something that combines all of those elements of a bergère (exposed wooden frame, closed back and arms, upholstery, generous comfy seat cushion, matching ottoman) and yet looks like a totally 21st-century chair.