1. A hackney-coachman. Now frequently applied to the driver of an Irish car.
2. A hackney-coach or jaunting car.
Pace Wikipedia, 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. It was the typical conveyance for persons in Ireland at one time (hence the reference by the OED to an "Irish car").
The Hackney Carriage (forerunner of the Hansom Cab, pictured) was first regulated in in London in 1654.
The OED gives the first use of jarvey (in the meaning of a coachman) in 1796. It is thought to derive as a by-form from the personal name Jarvis or Jervis.
Oh and forget the use of jarvey in Harry Potter. That's just part of the fiction.