A barouche was a four-wheeled carriage pulled by two horses with two double seats facing one another in the interior.
The barouche box was the raised seat for the driver.
To travel by post meant to travel in a hired carriage (similar to a diligence), usually with strangers. It was the public transport of the day. These hired carriages had fixed places to change horses, so they could make good speed.
Brighton is a town on the coast of East Sussex, in South East England. In the early 19th century, the Prince Regent commissioned the Royal Pavilion in Brighton. Many militia camps were held there as the Prince Regent reviewed them himself. By the end of the 18th century, it had become a fashionable sea resort.