C.E. Brock: illustration for the 1895 edition of "Pride and Prejudice"

A dogcart, or donkey cart, as it was called: the conventional mode of transportation for the Austens at the time, drawn by a single horse, with seating for two, sitting over two wheels. Larger versions could seat four, with passengers facing forward and backwards. The dogcart was soon upgraded to the dogcart phaeton, which was four-wheeled. Horses were expensive, of course, so these modes of transportation were signifiers of wealth.

The Austens' dogcart on display at Jane Austen House Museum (Chawton Cottage).

If you dive into the rapidly evolving technology of horse-drawn transport, an earlier form was the gig, which dates back to around 1790. They were also very popular in Austen's time, but they only carried one person. The Austens would have needed a two-seater.

If you wanted a racing model, and many wealthy young men did, the curricle (shown in the painting below) was for you. They rely on two horses and drivers often went too fast and there were lots of accidents. They pop up in some of Austen's novels.

John Cordrey: "A Gentleman with His Pair of Bays Harnessed to a Curricle" (1806)