The word beignet (pronounced bey-YAY) comes from the early Celtic word bigne meaning "to raise." It is also French for "fritter." Beignets, a New Orleans specialty, are fried, raised pieces of yeast dough, usually about 2 inches in diameter or 2 inches square. After being fried, they are sprinkled with sugar or coated with various icings.
Beignets are like a sweet doughnut, but the beignet is square shaped and without a hole. Beignets are considered the forerunners of the raised doughnut. When you hear people in New Orleans say, "Goin' fo' coffee an' doughnuts," they mean coffee and beignets. Beignet is one of the most universally recognized names for fried dough desserts which are basically fritter batter. For many years, beignets were shaped into balls or squares and covered with mocha frosting. Later the beignet was cut in the shape of a doughnut, and the raised doughnut was born.
I guess we called them doughnuts now..recipe can be found here