"Who invented the first motorcycle?" It seems like a simple question, but the answer is a bit complicated.

Motorcycles are descended from the "safety" bicycle, bicycles with front and rear wheels of the same size, with a pedal crank mechanism to drive the rear wheel. Those bicycles, in turn were descended from high-wheel bicycles. The high-wheelers were descended from an early type of push-bike, without pedals, propelled by the rider's feet pushing against the ground. These appeared around 1800, used iron-banded wagon wheels, and were called "bone-crushers," both for their jarring ride, and their tendency to toss their riders.

Daimler's wooden-framed "bone crusher"

Gottlieb Daimler (who later teamed up with Karl Benz to form the Daimler-Benz Corporation) is credited with building the first motorcycle in 1885, one wheel in the front and one in the back, although it had a smaller spring-loaded outrigger wheel on each side. It was constructed mostly of wood, with the wheels being of the iron-banded wooden-spoked wagon-type, definitely a "bone-crusher" chassis.

S. H. Roper's 1869 Steam-cycle

It was indeed powered by a single-cylinder Otto-cycle engine, and may have had a spray-type carburetor. (Daimler's assistant, Wilhelm Maybach was working on the invention of the spray carburetor at the time). If one counts two wheels with steam propulsion as being a motorcycle, then the first one may have been American. One such machine was demonstrated at fairs and circuses in the eastern US in 1867, built by one Sylvester Howard Roper of Roxbury, Massachusetts. There is an existing example of a Roper machine, dated 1869. It's powered by a charcoal-fired two-cylinder engine, whose connecting rods directly drive a crank on the rear wheel. This machine predates the invention of the safety bicycle by many years, so its chassis is also based on the "bone-crusher" bike.

The 5-cylinder Millet of 1892

Most of the development during this earliest of eras concentrated on three and four-wheeled designs, since it was complex enough to get the machines running without having to worry about them falling over. The next really notable two-wheeler was the Millet of 1892. It used a 5-cylinder engine built as the hub of its rear wheel. The cylinders rotated with the wheel, and its crankshaft constituted the rear axle.

The Mother of all motorcycle engines - the DeDion-Buton

The first really successful production two-wheeler though, was the Hildebrand & Wolfmueller, patented in Munich in 1894. It had a step-through frame, with its fuel tank mounted on the downtube. The engine was a parallel-twin, mounted low on the frame, with its cylinders going fore-and-aft. The connecting rods connected directly to a crank on the rear axle, and instead of using heavy flywheels for energy storage between cylinder-firing, it used a pair of stout elastic bands, one on each side outboard of the cylinders, to help out on the compression strokes. It was water-cooled, and had a water tank/radiator built into the top of the rear fender.

In 1895, the French firm of DeDion-Buton built an engine that was to make the mass production and common use of motorcycles possible. It was a small, light, high revving four-stroke single, and used battery-and-coil ignition, doing away with the troublesome hot-tube. Bore and stroke figures of 50mm by 70mm gave a displacement of 138cc. A total loss lubrication system was employed to drip oil into the crankcase through a metering valve, which then sloshed around to lubricate and cool components before dumping it on the ground via a breather. DeDion-Buton used this 1/2 horsepower powerplant in roadgoing trikes, but the engine was copied and used by everybody, including Indian and Harley-Davidson in the U.S.

First American production motorcycle - 1898 Orient-Aster

Although a gentleman named Pennington built some machines around 1895 (it's uncertain whether any of them actually ran), the first US production motorcycle was the Orient-Aster, built by the Metz Company in Waltham, Massachusetts in 1898. It used an Aster engine that was a French-built copy of the DeDion-Buton, and predated Indian (1901) by three years, and Harley-Davidson (1902) by four.