The origin of the doughnut is shrouded in mystery, but it probably began in mid-Nineteenth Century America when a woman named Elizabeth Gregory made pastries, each of which had a single nut in the center.
These were doughnuts and not, I should point out, donuts. The alternate spelling "donut" may have been the work of the restaurant chain Dunkin' Donuts. Kate Taylor of Business Insider summarizes linguistic evidence that suggests that the popularity of the spelling "donut" follows the rise of Dunkin' Donuts as a franchise empire.
After Dunkin' Donuts began in 1950, the use of the alternate spelling of doughnut emerged. It tracks with the spread of the company across America. The spelling has become so common that the Merriam-Webster Dictionary now lists donut has a correct way to spell the word.
-via Ed Driscoll