The first motorcycle?

"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 had no steering mechanism. Because it was impossible to steer around holes and bumps, they were called "bone-crushers". Only after the invention of the steer by Count von Drais, the push-bikes became somewhat more comfortable.

  
Gottlieb Daimler (who later teamed up with Karl Benz to form the Daimler-Benz Corporation) is credited with building the first motorcycle in 1885. It had one wheel in the front and one in the back, both of equal size, and a smaller outrigger wheel on each side to prevent tipping over. It was constructed mostly of wood, with the wheels being of the iron-banded wooden-spoked wagon-type, definitely a "bone-crusher" chassis.
It was powered by a single-cylinder Otto-cycle engine. In the first model it had an evaporation carburetor, which probably was replaced later by a spray-type carburetor which was invented by Daimler's assistant, Wilhelm Maybach.
 

If one counts two wheels with steam propulsion as being a motorcycle, then the first one may have been American. Around 1867 one Sylvester Howard Roper of Roxbury demonstrated such a machine, powered by steam, with 2 wheels and a steer at fairs and circuses in the eastern US.
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. Looking at the steel wheels and the rigid chassis, this machine must also be seen as a "bone-crusher" bike.

   
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 real attempt of building a motorcycle dates from 1892. De Millet used a 5-cylinder "star" engine built as the hub of its rear wheel. The cylinders rotated with the wheel, and its crankshaft constituted the rear axle.
  

The first real usable motorcycle comes from Hildebrand and Wolfmueller, who registered the motorcycle brand with the same name in 1894 at the Chamber of Commerce in Munich. They were also the ones who set up the first motorcycle production line.
As engine they used a parallel-twin, with direct connecting rods to a crank on the rear axle. New was the replacement of the heavy flywheel by a pair of stout elastic bands, one on each side outboard of the cylinders, to help to push the pistons back to their starting position. 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-Bouton built small, compact internal combustion engines in series.
In the first years the company built 1 type four-stroke engine of 138 cc. Bore and stroke were 50 by 70 mm. These engines were not really friendly for the environment. They used a so-called total loss lubrication system. From a tank oil dripped  through a metering valve onto the crankshaft bearings, which then sloshed it around to lubricate and cool components. After that it simply run through a hole out of the engine onto the ground.
The DeDion-Bouton company built its own tricycle equipped with this 1/2 horsepower powerplant and although they built more than 5000 of them, this engine was the most copied engine around the end of the 19th century. Most famous copiers are Indian (1901) and Harley-Davidson (1902).