A four-wheeled pleasure carriage, (commonly two-seated) somewhat like a phaeton, but having a straight bottom.
a light four-wheeled horse-drawn carriage; has two or four seats
A surrey is a type of 19th-century carriage, as celebrated in the Rogers and Hammerstein song The Surrey With the Fringe On Top; and also a colloquial expression for a four-seat vehicle related to a bicycle, stemming from the older usage.