The carriage shown here was manufactured by H.A. Moyer, of Syracuse, New York, early in the 1900s. The surrey is certainly a family-type vehicle, carrying 4 people easily, with an abundance of leg-room. American companies began building carriages of this type some time around 1870; however, those early versions did not have the graceful look of this fine carriage. The earliest 4 passenger Surreys had a straight top-line and one of the front seats would likely have lifted up to allow passengers access to the rear seat.
This extension top is foldable and settles down nicely and out of the way, behind the rear seat. Tom Ryder reminds us that these tops are often erroneously called "auto-tops" in the mistaken idea that they were the inventions of the early auto industry. Not so, they can be found on carriages 50 years before the first auto came along.
Berkebile* points out that the origin of the name is not known, though it is likely somehow associated with the English county of Surrey.
*Berkebile, Don H:
CARRIAGE TERMINOLOGY: AN HISTORICAL DICTIONARY [Smithsonian Institute]