The true doughnut, as well as the beignet, is deep-fried, as stated by PoppaGator. The name first appeared in 1803, listed as 'American' in a British paper, but the mention by Washington Irving, 1809, is usually cited. A Dutch origin is possible (see online articles).

There is a baked 'doughnut,' sometimes seen in supermarket bakeries, but it lacks flavor, character, presence, and doesn't deserve mention.

Here is a beignet recipe, PoppaGator may have a better one:

1 1/2 cups warm water
Dissolve 1 envelope active drt yeast in the water;
let stand 5 minutes.
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1/2 cup sugar, fine granulated
1 teaspoon salt (or less)
2 eggs, beaten
1 cup evaporated milk
7 cups flour
1/4 cup shortening (softened)

Oil (lard best)... for the deep frying

Powdered sugar
Touches of cinnamon, or... (optional)

Add sugar, salt, beaten egg and evaporated milk to the water-yeast mixture and whisk or blend with mixer. Add 4 cups of the flour and beat till smooth. Add shortening and slowly mix in rest of flour.
Cover and chill overnight.
Roll out on floured board, 1/8 inch thick. Cut into 3-inch squares.
Deep fry for 2-3 minutes at 360 F. (until lightly browned).

Drain and sprinkle with powdered sugar.