In Sense and Sensibility, we read:
... it would have quieted [Mrs. Dashwood's] ambition to see [Edward] driving a barouche. But Edward had no turn for great men or barouches. All his wishes centered in domestic comfort and the quiet of private life. Fortunately he had a younger brother who was more promising.
I've watched the Ang Lee version of that book multiple times, and in that version this information is conveyed by Edward Ferrars (Hugh Grant) lamenting to Elinor Dashwood (Emma Thompson, who also wrote the screenplay) that his mother wishes to see him driving a barouche, but that's not his desire. Perhaps it was the sound of the word "barouche" that caught my attention, but I've always remembered the word.
Wikipedia's definition: "A barouche is a large, open, four-wheeled carriage, both heavy and luxurious, drawn by two horses."
A couple nights ago, my friend Sarah (with thanks for her considerable assistance with this research) and I rewatched the 1995 version of "Pride and Prejudice," we both became fascinated at a very short (a couple seconds) scene where the roof of the Gardiner's carriage is folded down to convert it to an open carriage. I was vaguely aware that many carriages could be so converted, but this one had a hard top and several glass windows at the front rather than a collapsible leather cover. To a former mechanical engineer, that made the mechanism intriguing.
This led to both of us losing several hours of time down the rabbit hole of "carriages." The slightest mention (or sighting) in a movie or book can lead to fascinating discoveries. Horse-drawn carriages were a major form of transportation for centuries, and so there were many, many types and a huge amount written about them. Does knowing about them matter these days? In a practical sense, possibly only to the Amish, the British Royal Family (ie. people who still ride in them), and fans of period literature. But from a historical (and mechanical) perspective, it's fascinating.
What follows is a short list of carriage types. This is neither exclusive to Austen (I've included a couple out of curiosity that aren't in Austen) nor remotely complete (the number of carriage types over the centuries is huge). Definitions are from Wikipedia.
Barouche: "A barouche is a large, open, four-wheeled carriage, both heavy and luxurious, drawn by two horses." https://en.wikipedia.org/wiki/Barouche
Brougham: "A brougham was a light, four-wheeled horse-drawn carriage built in the 19th century." https://en.wikipedia.org/wiki/Brougham_(carriage) First built in 1838 or 1839, these weren't around in Austen's time.
Cabriolet: "A cabriolet is a light horse-drawn vehicle, with two wheels and a single horse. The carriage has a folding hood that can cover its two occupants, one of whom is the driver." https://en.wikipedia.org/wiki/Cabriolet_(carriage)
Chaise: "A chaise, sometimes called chay or shay, is a light two- or four-wheeled traveling or pleasure carriage for one or two people with a folding hood or calash top." https://en.wikipedia.org/wiki/Chaise
Four-in-hand: "A Four-in-hand is any vehicle drawn by four horses driven by one person. Driving large heavy carriages and private coaches drawn by four horses was a popular sporting activity of the rich after the middle of the 19th century." https://en.wikipedia.org/wiki/Four-in-hand_(carriage) I doubt this appears anywhere in Austen as it appears to post-date her.
Gig: "A gig, also called chair or chaise, is a light, two-wheeled sprung cart pulled by one horse." https://en.wikipedia.org/wiki/Gig_(carriage)
Hansom: "The hansom cab is a kind of horse-drawn carriage designed and patented in 1834 by Joseph Hansom, an architect from York. ... Originally called the Hansom safety cab, it was designed to combine speed with safety, with a low centre of gravity for safe cornering." https://en.wikipedia.org/wiki/Hansom_cab Again, this post-dates Austen - but it's particularly interesting because Sherlock Holmes took these a lot.
Landau: "a landau is a four-wheeled carriage with a roof that can be let down. It was a luxury carriage." https://en.wikipedia.org/wiki/Landau_(carriage)
Landaulette: "A landaulet or landaulette carriage is a cut-down (coupé) version of a landau horse-drawn carriage. The landaulette retains the rear half of the landau's two-part folding top." https://en.wikipedia.org/wiki/Landaulet_(carriage)
Post chaise: "A post-chaise is a fast carriage for traveling post built in the 18th and early 19th centuries. It usually had a closed body on four wheels, sat two to four persons, and was drawn by two or four horses." https://en.wikipedia.org/wiki/Post_chaise
Victoria: "The victoria is an elegant style of doorless four-wheeled open carriage, drawn by one or two horses, based on the phaeton with the addition of a coachman's seat at the front, and with a retractable roof over the passenger bench." https://en.wikipedia.org/wiki/Victoria_(carriage) This would also have post-dated Austen.
The carriage in the 1995 "Pride and Prejudice" turned out to be a "Landau," but not just any Landau: it's an uncommon "Five Glass Landau," named for its abundance of windows. ( https://www.thecarriagefoundation.org.uk/item/five-glass-landau ): "The Five Glass Landau is an unusual carriage which instead of having two leather heads in the classic Landau style has only a rear head with the front one being replaced by three windows, which can be dismantled and stowed away, thus creating a very open carriage when one was required.” Not only could the glass be removed, but the entire top would then collapse and fold under the driver's seat.
Here's a blog with photos of the actual carriage used in the 1995 mini-series, from Red House Stables & Carriage Museum, Derbyshire:
Even if you understand all these carriage types, then you have to get into the associated terminology - and there's a huge amount of that. For example: "A postilion rode on the near-side (left, nearest the roadside) horse of a pair or of one of the pairs attached to the post-chaise leaving passengers a clear view of the road ahead." It's not clear to me, but I'm pretty sure that makes the "postilion" what we would consider the "driver."
As a side note ... Ironically, the "Landau Bar," the elongated S-shaped bar used to help collapse the roof, became such a symbol of wealth that it could be seen on the side of the cabin of multiple cars as a non-functional item into the 1990s. ( https://en.wikipedia.org/wiki/Landau_(automobile)#Landau_bar )
If this subject interests you - particularly for fans of Austen - it's worth reading this JASNA (Jane Austen Society of North America) page and watching its two videos (about carriages in Austen):