A two-wheeled horse-drawn vehicle is a cart (see various types below, both for carrying people and for goods). Four-wheeled vehicles have many names – one for heavy loads is most commonly called a wagon.
Very light carts and wagons can also be pulled by donkeys (much smaller than horses), ponies or mules. Other smaller animals are occasionally used, such as large dogs, llamas and goats (see draught animals).
Vehicles pulled by one animal (or by animals in tandem – single file) have two shafts which attach either side of the rearmost animal (the wheel animal or wheeler). Vehicles pulled by a pair (or by a team of several pairs) have a pole which attaches between the wheel pair. Other arrangements are also possible, for example three or more abreast (a troika), a wheel pair with a single lead animal (a "unicorn"), or a wheel pair with three lead animals abreast (a "pickaxe"). Very heavy loads sometimes had an additional team behind to slow the vehicle down steep hills. Sometimes at a steep hill with frequent traffic such a team would be hired to passing wagons to help them up or down the hill.
Two-wheeled vehicles are balanced by the distribution of weight of the load (driver, passengers and goods) over the axle, and then held level by the animal – this means that the shafts (or sometimes a pole for two animals) must be fixed rigidly to the vehicle's body. Four-wheeled vehicles remain level on their own, and so the shafts or pole are hinged vertically, allowing them to rise and fall with the movement of the animals. A four-wheeled vehicle is also steered by the shafts or pole, which are attached to the front axle; this swivels on a turntable or "fifth wheel" beneath the vehicle.
Vehicles primarily for carrying people
A horse and buggy circa 1910
Road
Ambulance: Much the same purpose as the modern sense. Details of the design varied but would be a lightly-built and well-sprung, enclosed vehicle with provision for seated casualties and stretchers.
Barouche: An elegant, high-slung, open carriage with a seat in the rear of the body and a raised bench at the front for the driver, a servant.
Buggy: A light, open, four-wheeled carriage, often driven by its owner. It is an American design.
Cab: A shortening of cabriolet. Joseph Hansom based the design of his public hire vehicle on the cabriolet so the name cab stuck to vehicles for public hire.
Traveling in France or Le départ de la diligence
Drawing by George Cruikshank (1818).
Carriage: In the late eighteenth century, roughly equivalent to the modern word "vehicle" [Walker]. It later came to be restricted to "passenger vehicle" and even to "private, enclosed passenger vehicle" [Britannica]. This last is the sense adopted by the linked article.
Covered wagon: The name given to canvas-topped farm wagons used by North American settlers to move both their families and household goods westward. Also called Conestoga wagon and prairie schooner.
Ralli car: a light two wheeled sprung cart (gig) with two forward-facing and two rear-facing seats back-to-back, and a sliding fore-and-aft seat adjustment to allow the vehicle to balance with different numbers of passengers.
Training cart or training trap: A simple sprung or unsprung two-person modern cart for training a harness horse on smooth roads. Often made of steel with motorcycle wheels, and sometimes with adjustable shafts for different-sized horses.
Trap: An open sprung cart. Often used in a general sense to cover any small passenger-carrying cart.
Troika: A sleigh drawn by three horses harnessed abreast. Occasionally, a similar wheeled vehicle.
Vardo (gypsy wagon): A vardo is a traditional horse-drawn wagon used by English Romani Gypsies.
Victoria: A one-horse carriage with a front-facing bench seat. The body was slung low, in front of the back axle. Driven by a servant.
Fly boat: A canal boat which changed horses at stages and could therefore keep moving, care being taken to maximize its speed.
A basic, un-sprung cart in Australia. In that country and in New Zealand, it is known as a dray (but "dray" elsewhere usually means a four-wheeled wagon).
Vehicles primarily for carrying goods
Road
Bow wagon: A simple agricultural wagon with laths bowed over the wheels in the manner of mudguards, to keep bulky loads such as straw from contact with them. An Australian design.
Un-sprung cart: A simple two-wheeled vehicle for workaday use in carrying bulk loads. It was usually drawn by one horse.
Chasse-marée: A four-horse adaptation of the cart principle for the rapid delivery of fish to French markets.
Conestoga wagon: A large, curved-bottom wagon for carrying commercial or government freight. See covered wagon.
Dray: Particularly in Australia and New Zealand, an un-sprung cart. In Britain, even in the 18th century, the name came to be associated with brewers' deliveries so that the later vehicle that was more correctly called a trolley also came to be known as a brewer's dray. These are still seen at horse shows in Britain.
Also a sledge used for moving felled trees in the same way as the wheeled skidder. (See implements, below). It could be used in woodland, apparently with or without snow, but was useful on frozen lakes and waterways. [OED]
Float: A light, two-wheeled domestic delivery vehicle with the centre of its axle cranked downward to allow low-loading and easy access to the goods. It was used particularly for milk delivery.
Lorry: A low-loading platform body with four small wheels mounted underneath it. The driver's seat was mounted on the headboard.
Mail coach: A stagecoach primarily for the carriage of mail, though also carrying passengers.
Mophrey: An un-sprung cart which could be extended forwards with the addition of front wheels. It was used by small farmers as and when dense or bulky loads were to be carried (muck-spreading and harvest). An eastern English design.
Prairie schooner: The name given years later to the canvas-topped farm wagons used by North American settlers to move their families and capital goods westward. See covered wagon and Conestoga wagon.
Travois: A very simple sledge used for moving relatively small loads, consisting of a pair of shafts dragging on the ground.
Trolley: Like a lorry, but with slightly larger wheels and slightly higher deck. The driver's seat was mounted on the headboard.
Trolley and lift van: A standardized trolley and a lift van, a standardized box, designed to fit each other or any other of the same sort. The lift van was the direct counterpart of the modern container in the materials and size appropriate to its time.