Of course, carriages can't go anywhere without horses, and like most modern people, I don't have a ton of experiences with the creatures, other than being overly obsessed with them as a little girl. So, here's some general information about horses that you might find interesting.
(or Coach Box)
Box Seat or Coach Box: the front seat where the coachman sits.
Hammer Cloth: decorative cover draped over the coachman’s seat (often fringed and bearing the family crest). Generally used in formal occasions.
Head: Covering used to protect the passengers from weather. Some carriages have a permanent, fixed head (such as the coach or chariot), though most have foldable ones that can be pushed down during fine weather. The style of head is one of the most distinguishing factors from carriage to carriage.
Footman’s Cushion: during ceremonial processions, this is where the footman would stand, holding onto a strap called a “Footman’s Holder”.
Boot: A compartment at the back of a carriage that could be used as seating for the servants and sometimes held a compartment for holding luggage or goods. In America, this was known as a “Rumble Seat”. They had special luggage used with carriages called "imperials." They were made of wood and covered in leather, then strapped to the head and / or boot.
Perch: the framework under the carriage that provides structure for the coach.
Spring: an essential part of the undercarriage. Mostly constructed from metal, some are also supported by leather straps (the older coaches).
Splinter Bar: The bar that goes across the front of the carriage that attaches to the horses. The horses then are attached to it by poles, shaftes, of swingletrees, and by leather traces that attach to their harnesses.
The earliest form of suspension involved hanging the carriage body on leather braces between two frames (the fore and hind standards). Advances in the manufacture of steep springs changed the appearance of carriages, allowing bodies to be slung lower, more compact, and cheaper to produce. The passengers got a more comfortable and safer ride.
One of the older styles of suspension. It allows the body of the carriage to swing and move freely. State carriages all use cee-springs, and it can cause motion sickness. At the National Trust Museum, they had a seat fitted with this suspension to give you the feel of what it felt like to ride with it, and it was super bouncy! It felt really delicate, and frankly, it felt like I couldn't put my full weight on it because it sunk so far down that it felt like it wouldn't hold me.
Elliptic Spring Effect
These made carriages lighter and less expensive. It made the suspension more solid and giving a more comfortable ride. The museum had a seat with this type as well, and it wasn't bad to sit on, though I can imagine that it still would be difficult to handle a long trip with only that to soften the rough roads.
The first braking systems were primitive. Originally, there wasn't really anything to slow the carriage. Basically, stopping the horses and letting the carriage roll to a stop on its own was the only option.
Then, they created "drag shoe" (or sometimes spelled "dragshoe"). It came in a variety of designs, but its function was to attach to the back wheel to slow it, kind of like an antique brake pad. Often, it was attached to the carriage on a chain and either the coachman or groom would jump out and attach it when they crested a hill and then removed it when they reached the bottom. Eventually, handbrakes were introduced that allowed the driver to operate the drag shoe.