A barouche was an upper-class type of carriage used in Georgian and Victorian England. During this period, there were an overwhelming number of different types of carriages and cabs, and each had a different message about the rider's social class and pocketbook. A barouche was primarily used for leisure driving in nice weather, and as such, advertised that the owner could afford an 'extra' carriage just for summer cruising -- and a nice carriage, at that. Because of this, barouches probably appear in Victorian era stories (e.g. Sense and Sensibility, Emma, or the Aubrey-Maturin series) more often than you might reasonably expect, to illustrate the wealth and class of certain characters. Barouches were such a mark of higher-quality people that the well-known Four-Horse Club was sometimes called the Barouche Club, although this was in part because the Club mandated the color, decoration, and type of horses that members should use with their barouches.
The classic barouche is a four-wheel carriage, with a set of double seats facing each other, so that the two riders in front were facing backwards. The driver sat up front in a raised seat, apart from the passengers. While the carriage was essentially open, there was a fold-up canopy that could be used to shade or shelter the rear-most seats. The carriage was pulled by a set of two horses.
There are a number of variations on the barouche. A light barouche, with only one bench seat, was called a barouchet (or barouchette). The lowrider version was called a barouche-sociable, allowing the passengers to chat comfortabally with pedestrians. A barouche with a second fold-up top for the front seat was called a barouche-landau. Of course there were many other types of carriages that looked something like a barouche; the landau, the victoria, and the phaeton, for example, are all close enough to the barouche that the modern American might not see the difference. On the other hand, a Victorian probably wouldn't be able to tell the difference between a Toyota Prius and a Honda Civic.
The name barouche is a misnomer, taken from the German Barutsche, which originally referred to a vehicle more akin to a cabriolet, having only two wheels. The word has its original root in the Latin birotus, literally meaning two-wheels. Barouche is pronounced 'buh-roosh'; this is apparently an attempt to make it sound French, and thus stylish.
In modern times, the barouche is one of the more common carriages found for hire in parks, and it is often used in weddings (including my own) if the bride wishes to make a spectacular entrance (or exit). These modern barouches tend to be a little more heavy-set than the classic design, but are still the mark of elegance.