The phaeton was a small, lightweight carriage, and not much to look at by today's standard of storybook carriages. But in the early 1800s it was one of the sports cars of the day. Not, admittedly, because of its horsepower, but because it was small and speedy and agile, and because only the wealthy could afford such an unnecessary carriage. It was pulled by only one or two horses, and intended to carry only one or two passengers (although some had a small seat for a groom in the rear. It was a bit of an oddity in that it was actually driven by the owner, rather than a coachman.
While there was a truly dizzying array of carriages labeled as phaetons, in general they all shared a comparatively light build, with open sides, no doors, large wheels, and small canopy to protect the driver from the sun or light rain. They were well-sprung, although they appeared around the same time as macadamization, thus making this feature less important in some models. Because they were fast and light they were comparatively dangerous and prone to tipping over, although there were models that were designed more appropriately for ladies and the elderly.
The phaeton was an interesting mix of the decadent and the dangerous, which is part of the reason for the large variety of makes and models. The popular spider phaeton was particularly light, with a small body and narrow wheels, fast and nimble. On the other end of the spectrum was the George IV Phaeton, designed for an elderly and rather fat King George, who had enjoyed the more sprightly phaetons of his youth and requested one that he could easily mount. This low-slung, less sporty design later became known as a 'park phaeton' or 'lady's phaeton'. If you wanted a more rugged phaeton, you might go for a mail phaeton, designed with the same spring system as the mail carriages, and often used by gentlemen on longer journeys. (Of course, a long journey called for a proper coach in most cases; the phaeton was used for travelling by post, moving quickly and stopping only to trade out hired post horses as each set tired). There were less practical models too, including the high perch phaeton, which was designed with such an impressive suspension system that you needed a ladder to mount it.
The phaeton was considered the youthful and dashing counterpart to the more respectable and stately landau. However, if you wished to be even more dashing and daring, one might opt for a curricle, a small two-wheeled chaise which also used two horses -- a truly overpowered mode of transportation, even faster and more dangerous than the phaeton.
The phaeton was named after the Greek myth of Phaethon, which literally means 'shining'; it most likely gained this name because classical references were fashionable in Europe at this time. It is pronounced fey-it-in.