A term of abuse hurled by street urchins of the time at the riders of what was, then, just "a bicycle", and is more properly known as an "ordinary" (the name given to the design patented by James Starley in 1871), a " high bicycle" or "high-wheeler", the latter being something of an Americanism; aficionados and collectors generally dislike and avoid the term "penny-farthing" to this day.
The earliest successful pedal-driven bicycles, as designed by Pierre Michaux, were driven by pedals and cranks attached directly to the axle of the front wheel. This means that the only variable that could be used to change the distance travelled per pedal revolution - the gear, in other words - was the size of the wheel. Thus, with riders wishing to be able to go faster and pedalling speed reaching its physiological limits, the only option was to increase the size of the wheel. As the front wheel got larger eventually the limiting factor, the length of the rider's legs, meant that he (or, after the invention of rational dress, she) ended up perched above the front wheel rather than between the front and the rear. The largest wheel sizes generally used were around 56 inches in diameter. (One hangover from these days is the peculiar English-speakers' habit of quoting contemporary bicycle gearing in terms of the equivalent wheel size of a direct drive machine, so that a 52 tooth chainring, a 13 tooth sprocket (a 4:1 drive) and 27" wheels gives you a "108 inch gear", ie the equivalent of riding a high bicycle with a wheel 9 feet in diameter. But I digress.)
The wheel size gave some advantages when tackling the bumps on the unmade roads of the era, but as the rider's position meant that the centre of gravity of the machine was only marginally behind the front hub, about which the whole rider/machine assembly could pivot, effective braking was impossible, since any serious deceleration would put the rider over the handlebars; a spoon brake operating on the running surface of the tyre was nonetheless generally fitted. When descending hills, riders would put their legs over the handlebars so as to be able to fall free of the machine and, hopefully, land feet first in the event of a tumble. Mounting was generally accomplished by running alongside the bike to get it up to speed and scooting on the step a couple of feet up the frame member before swinging yourself into the saddle. Although the rider's elevation seems fairly terrifying to those familiar with modern bikes, and is indeed quite an uncomfortable height to fall from, it is of course very similar to the position of a rider on horseback, which would have been familiar territory to many more people in the 19th century than it is today.
Riding one of these things was, evidently, something for the more athletically - or indeed, acrobatically - inclined. For those of a more sedate disposition, manufacturers dreamed up a wide range of designs for tricycles, quadricycles and at least one five-wheeled monstrosity (known to the same urchins as the hen and chickens for its one large central wheel and four small ones; they were all, however, unwieldy and expensive. The sport of cycle racing really took off in this era (although the first road races dated back to Michaux's machines of the 1860s) and some prodigious feats were performed: in 1886, G.P. Mills set the first official British "end to end" (Lands End to John O'Groats) record, covering what was then 900 miles on unmade, unlit roads in five days, one hour and 45 minutes.
However, by the time of Mills' epic ride the writing was already on the wall for the Ordinary. James Starley's nephew, John, had started production of the "safety bicycle" with a chain-driven rear wheel - essentially identical to the modern bicycle in form; although racers initially scorned the Safety, Mills soon rode one to beat his own end-to-end record by more than half a day. Soon the old high bicycle had almost vanished from the scene, and freed from the morphological constraints of leg length and wheel size along with much of the danger of falling, cycling's real boom era took off.