What is a moped really and where does it come from?
To answer this question, one needs to go back to the early days of the bicycle and the invention that revolutionised the 20th century - the internal combustion engine. Put one of these into a coach and you have an automobile; mount it on a bicycle and presto, you have a moped (MOtor+PEDals), the precursor of ALL motorcycles. The pedals were omnipresent on all, used both as a starter device and as emergency fallback on human power. As engine sizes got bigger and bigger, it looked as if the half-bicycle-half motorcycle mopeds were just a short-lived early development phase (like a tadpole) that progress simply left behind. History however proved it otherwise.
The moped's evolution can be broken down into 4 distinct periods: from the very beginning to the end of the first world war, the period between the two wars, the phase after WWII to the early '80s and finally today. We also have to define what do we call a moped: a cross between a human-powered bicycle and an engine powered motorcycle, equipped with a (usually) under-50cc gasoline engine (although some early models used 98cc displacements as well).
After the end of WWI, with Europe's map completely redrawn, the short period until the Great Depression of the '30s was not conducive to the developemnt of the moped. While motorcycle manufacturing boomed, largely fuelled by the rapid re-armament needs on all sides, it seemed that the low-cubic, low-speed moped would become extinct forever.
Except nobody factored in WWII. After the war ended in Europe, the demand for simple and economical (read: cheap and affordable), means of transportation skyrocketed. Nobody could afford a car, or even a motorcycle. Most factories were in ruins anyway. In Southern European countries, like Italy and France, people got around on bicycles. This was fertile breeding ground for the rebirth of the moped and also gave us the other Italian cultural icon, the scooter. Bicycle makers (most of which were also motorcycle manufacturers) all started offering small auxiliary engines for their bicycles. Motobecane, Peugeot, Ducati, Moto Guzzi all got into the act.
The first giant leap forward was the appearance of the Velosolex, a giant among the dwarfs. If you ever rode a Velo, either with its engine running or as a bicycle, you'll appreciate how well that entire machine was constructed. Originally equipped with a 33cc engine (later to grow to 42 then 49) it had a friction roller over the front wheel. You could engage-disengage the power with a lever. You had to help sometimes going uphill, but that was part of the experience.
The Solex became a genuine cult idol, and rightly so. Amazingly, after more than 50 years, it'still going strong.
The common ancestor to mopeds and motorcycles; the Benz 'rolling saddle' from 1886. Still with training wheels and lever-type pedals.
A Harley from 1905. Srtill with pedals.
The scene changed dramatically, when Steyr-Puch of Austria, the company founded by Johann Puch, a master bicycle maker in the late 1800's, introduced the first MS-50 in 1952. It represented a dramatic departure from the bicycle-based design principles; with its steel pressed frame, fan-boosted engine cooling, 2-speed handlebar-shifted gearbox and an elaborate electrical system it had more in common with 'real' motorcycles; the first moped that was designed to be driven by its engine most of the time.
The MS-50 design's success attracted many followers and became the kingpin of its era. It remained in production virtually unchanged until 1982. They reached North-America in the late '50s through the Sears catalogue, as the Allstate Mo-Ped.
The 1947 prototype Velosolex, virtually unchanged ever since and still going strong
The 1955 MS-50. Notice the long wheelbase, and the characteristic engine cowling.
Shock absorbers front and rear, detachable tank, big headlight and 19" wheels. This 1959 Allstate Mo_Ped could go 60 kmh (40 mph). You could order it from the Sears catalogue.

As Europe gradually recovered economically, again it looked that the moped, originally intended as a stopgap measure would disappear, this time for good. Instead, it kept on flourishing and its popularity went on to become a craze.There were moped races, clubs, meets and trips.
And for those, who belittle a 50cc 2-stroke engine as 'puny' here's a piece of historical trivia: the world speed record in the 50cc class was set in 1965 by a 49cc Kreidler stock moped (tuned and in an aerodynamic enclosure), at 223 kmh. WOW.

Mopeds were (and still are) typically powered by a 2-stroke engine, although some experimented with 4-stroke (Ducati, Honda, Indian, Motom)
The 1959 Eysink Credette from Holland. Notice the curvacious shape and the integrated tank.
Below is one of the iterations of the moped concept, the Tandem. There were also moped tricycles, mopeds with sidecar, or utility mopeds with a trailer-like big rolling storage bin up front.
The problem with 4-stroke engines in the under-50cc class is that they do not produce much power (very low torque at low RPM's) therefore the acceleration (the ability to take off fast enough from standing still not to hold up traffic) is fairly poor. 2-stroke engines can be tuned to much higher power outputs, using refined port timing, reed valves and high grade oil in the micxture; this is why their popularity still holds today (the other reason is that in many countries under 50cc engines are exempted from emission controls).

This beautiful 1980 Indian carries the famous brand' name and was meticulously conceived as the 'non-plus'ultra' among mopeds. With its large frame, extended handlebars, elongated seat (sold as a 2-seater where permissible) riding one didn't feel like riding a smaller, more bicycle-like moped.

By the late '60s mopeds branched off into multi-speed and cheaper single speed versions. Other than the 2-speed manually shifted Puch MS-50 and its cousins, 4 and 5-speed versions appeared as well . Soon the auto-shift 2-speed models arrived, followed by the continuous variable transmissions (variators), the same system that is commonly used today on virtually all modern scooters.
By the late '70s, the field was crowded with hundreds of styles, models and brands; the only common trait was the under-50cc engine size. Worldwide sales were in the millions. Even in North-America, that only caught the tail end of the boom, mopeds sold in spectacular numbers. In 1975, in the US alone, some 125 different models were available; Canada had about 25. They were sold through car dealerships, bicycle shops, county fairs, hardware stores, garden supply places (together with lawnmowers).
The boom continued unabated until the early 1980s. But then, the worldwide recession hit and hit hard. Many motorcycle and bicycle makers worldwide were wiped out. This, combined with the introduction of mandatory licensing and insurance signalled the end of the moped craze as we know it.
For the third time it seemed the moped was finished. And for the third time, it bounced back.
In 1997, close to 12 million were produced world wide. And while the distinction between 'true-blue' mopeds (with pedals) and their siblings without ('no-peds') and even scooters gets blurred (watch the emergence of the 'cyclo-scooter'), the staying power of small engine-powered 2-wheelers is now undisputed.

Mopeds are here to stay.

The latest addition to the moped class is the TOMOS Revival (Renaissance) a stylish, crusier-type moped with lost of chrome, extended wheelbase, large weigth carrying capacity, electric start.