A gig is an elegant 2-wheeled cart used by the well-to-do (originally it actually was considered a cheap carriage, but by the 19th century,the design and manufacture had improved so much it was the vehicle of the wealthy gentleman who wanted a convenient way to transport himself and one other person). It was usually built quite high off the ground and with curving shafts. It is pulled by a flashy horse. There are quite a few different styles of gigs, all named after their amnufacturers, and as society became more mannered, certain styles got associated with male or female drivers. The Stanhope gig and Dennett were driven by men. Women could drive the Tilbury.
As the gig tended to be heavy and pulled by a flashy animal, who was often a handful, it tended to be more of a gentleman's vehicle. Ladies would drive govenerness carts or dog carts, which were easier to hop in and out of in an emergency.