Count Rumford, for whom the fireplace is named, was born Benjamin Thompson in Woburn, Massachusetts in 1753 and, because he was a loyalist, he left (abruptly) with the British in 1776. He spent much of his life as an employee of the Bavarian government where he received his title, "Count of the Holy Roman Empire." Rumford is known primarily for the work he did on the nature of heat.
Rumford fireplaces were common from 1796, when Count Rumford first wrote about them, until about 1850. Jefferson had them built at Monticello, and Thoreau listed them among the modern conveniences that everyone took for granted. There are still many original Rumford fireplaces -- often buried behind newer renovations --throughout the country.
Ah, American ingenuity and how quickly we forget it.
The This Old House website has a quick primer on what makes a Rumford different:
In a traditional fireplace, the fireback slopes forward to direct smoke up the chimney. Incoming air spills over the sharp edge of a steel lintel eight or nine inches below the damper, mixes turbulently with the smoke and "rolls" upward. While this construction prevents the fireplace from smoking, it also loses some of the fire's heat up the chimney.
Count Rumford designed a fireplace with a high, wide opening, a shallow firebox and widely splayed jambs to reflect as much radiant heat out into the room as possible. But Rumford's real genius was straightening the fireback and rounding the front wall of the throat, essentially creating a nozzle—like an inverted carburetor—that shoots smoke up through the damper and out the chimney, wasting less heat in the process.