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 hat is one of the more sane and sensible applications of the word, as it is, in its base form, a quite sensible and useful hat.
A bergère is a flat, wide straw hat, with a very low crown and a very wide brim. It is, in essence, the closest thing you can get to wearing a plate on your head without looking entirely unlike a hat. Its purpose was to function as a sunshade, and as such it was traditionally made of woven straw to allow for good airflow.
Then came the French fashion world's obsession with faux-peasant aesthetic, and suddenly the bergère found itself silk-lined and covered in embroidery, ruffles, and fake flowers. First rising to popularity in the 1730s, they were revived in the 1860s, and then in later decades had close descendants in the form of Gainsborough hat (AKA picture hat), although these often had quite large crowns to carry more finery and feathers.
While there is a fair amount of variation, classically a bergère hat would be an unadorned straw hat, or have just enough of a crown to tie a ribbon around. The hat would be affixed to the head by means of a wide ribbon sewn into the hat and tied under the chin. Over the decades, most variations you might imagine have been constructed, with the most common changes being the use of fancier materials or fancy decorations; however, a surprising number of the fancier designs continued to use the basic straw hat for their foundations, presumably in an attempt to maintain the reference to its rustic roots.