Patented in 1834, the hansom cab was a two-wheeled cabriolet for two passengers, with a driver seated behind with the reins going over the roof. These were typically public vehicles for hire, but many were privately owned and considered rather dashing such that no lady would venture out in one alone. Road Cart
A cart is typically a two-wheeled wagon that was maneuverable and drawn by a single horse. It was a general-purpose trade or farm vehicle with no suspension.
Stanhope Gig
A feature of the design was that the shafts were stiffened throughout their length by a steel plate fitted to the undersides and attached directly to the axle by two turned pillars at each side. The body was suspended by a four-spring combination of two side- and two cross-springs. This combination became known as “Stanhope springs.”  
A jaunting car is a light two-wheeled carriage for a single horse, in its most common form with seats for two or four persons placed back to back, with the foot-boards projecting over the wheels.
There were two main varieties of jaunting car: the "outside jaunting car", or "outside car", the more common type described above, in which the passengers faced outward over the wheels, and the "inside jaunting car", or "inside car", considered to be more "genteel", in which the passengers sat with their backs to the sides of the car and faced each other.
       
       
       
Meadowbrook
This carriage originated in Long Island, NY. where it was known as an East Williston Cart. They have seatbacks that fold, a dashboard, and often times fenders. Meadowbrooks in general are considerably more comfortable than road carts. 
Gigs were light, two-wheeled, one-horsed vehicles for two passengers. This was the most common vehicle on the road.  A curricle was a smart, light two-wheeled chaise or "chariot", large enough for the driver and a passenger and— most unusual for a vehicle with a single axle—usually drawn by a carefully matched pair of horses.  This small, light, two-wheeled vehicle held 2 passengers and was drawn by a pony. This is often what is meant by a dog-cart when used improperly.