Barouche

The barouche was fashionable in the 19th century and often regarded as a fancy carriage. Four-wheeled and shallow, the vehicle had two bench seats inside, facing each other. Two people in the front faced two in the back. People would step up into the space between them to enter. A collapsible half-hood folded over the back seat. The driver sat up high in the front, outside of the vehicle on a seperate box. A pair of complementing or matching horses or multiple pairs drew barouches. These carriages were used mostly for summer leisure driving. There was a light barouche called a barouchet or barouchette.

Cabriolet

...a light two-wheeled one-horse carriage that had a folding leather top and upward-curving shafts. It seated two people. There was usually a rear platform for a groom. The vehicle was first used in France. It handled well and could be navigated on busy streets. It replaced the hackeny carriage as a vehicle for hire in Paris and London.
http://cabriolet

Tilbury

A tilbury is a light, open, two-wheeled carriage for two persons, developed in the early 19th century. The tilbury's large wheels could move fast over the rough roads. This vehicle was sporty and dangerous.

Dog Carts

For use in the country rather than in town, people would often use the "Dog Cart." It had two wheels. The driver sat on one seat, and two people could sit on another seat facing the rear.

Brougham carriage

A brougham was a light, horse-drawn vehicle of the 19th century. This carriage is entirely closed. Passengers entered through a center door on its side. It had four wheels. Two people could sit within it, facing each other. A box seat in the front was provided for the driver and a footman or other passenger. There was a glazed front window.

It was designed to be drawn by a single horse.
Upper class families used them during bad weather.
Electric Broughams came later.

Curricle: charming owner driven cart

Popular in the early 19th century (England, France, Italy) with fashionable young men who wanted to drive themselves was the curricle, a light, elegant, two-wheeled chaise or chariot. It could hold the driver and a passenger. This vehicle was drawn by a carefully-matched pair of horses tied together by an "I" shaped curricle bar.

Curricles were known for the accidents-if one horse fell, the other was likely to follow. It had a single axle, and a light dashboard hung with a couple of lamps.
By the mid 19th century, this vehicle began to lose popularity.

The Victoria (carriage)

Victorias are elegant, single-horse, coachman-driven vehicles. It was popular with the wealthy. The body is low, and it has a forward-facing seat for two people. The driver's seat is raised, and it has a calash top.