This one's for the illustrators. I've seen some very inaccurate depictions of horses in harness, so I'm going to pull up some visual aids here and talk about it.
There are several kinds of horse harness. One of the big differences is whether a collar is used or not - collars provide a more effective pull, but are heavy on horses that might be asked to go fast.
Image source: Amanda Slater via Wikimedia Commons.
So, let's start with this. These are plow horses, although they are wearing "show" harness - probably for a plowing contest. The weight of the plow is settled on the collars around the horse's neck. The horse collar is a very efficient and comfortable (for the horse) way to have an equine pull weight. The stiff framework supporting the collar is called the "hames". The plow itself is attached to chains which are connected to the sides of the collar. Chains are used because their weight helps them fall into place better than straps or ropes. The chains run through a strap around the horse's body, which in harness terms is the "saddle". A strap along the back supports britching, and there's also a strap between the horse's legs. All of this is to keep the saddle and collar from slipping. The decorated brass discs are called "horse brasses" and are an old British tradition. The blinkers help keep the horses from spooking and being distracted.
Image source: Rooh23 via Wikimedia Commons.
This is a horse pulling a cart in collar harness - you can see the same collar and a similar bridle, but this particular harness has no saddle. Instead, it has two straps over the back which support the shafts, which are then also secured to the collar. Again, the weight of the cart rests primarily on the collar. The rest is all to stop stuff moving and, in this case, to prevent the shafts from moving up and down.
Image source: Adam Jones via Wikimedia Commons.
A rather more typical horse-drawn carriage - note the short chains between the end of the shaft and the collar (which is also a much lighter collar). This harness does have the more typical collar. As this is a double, the single shaft runs between the horses - this is typical.
This is "breast" or "breast collar" harness - the weight of the cart is supported on a strap across the horse's chest. This kind of harness weighs less than a collar and is easier to put on, but isn't efficient or comfortable for heavy weight. In this picture, note the lack of breeching - instead, there's a crupper which runs under the horse's tail. Also note that both this team and the one above appear to be wearing bonnets - those are to keep the flies out of the horses' ears.
Either of these would be correct for a typical horse-drawn carriage pulled by a pair of horses.
Image source: Vickusin, via Wikimedia Commons.
This shows a four-in-hand in modern competition. Here, the shafts are on the outside of the back pair of horses, called the "wheel" pair. There are no shafts next to the front or "lead" pair. Note the strap holding the lead pair together - this is to keep them straight and at the correct distance from each other without shafts in the equation. The driver here also has two sets of reins or "lines" - one for each pair of horses. This allows him to turn the front pair separately from the rear pair when making tight turns. This sort of setup would be correct for a stage coach or large carriage with four (or more) horses.
Image source: Marjon Kruik via Wikimedia Commons.
A slightly different setup, also correct. The shaft is between the wheel pair and the horses are rigged out with breast collars rather than neck collars.
Image source: Benny Mazur via Wikimedia Commons.
When four horses isn't enough. A six-up of draft horses pulling a wagon - this would be a correct setup for large scale farm deliveries. The shaft runs between the second and third pairs of horses. (The middle pair is called the swing pair). Hitches of eight or even ten horses are known. Just add horsepower...
Image source: Quistnix via Wikimedia Commons.
This is a more unusual setup - a tandem of ponies. It also shows very classic britching very well on the first pony.
And another less common arrangement - a four abreast of Norwegian Fjords. Image source: Pete Markham via Wikimedia Commons.
Image source: Jon via Wikimedia Commons.
Today we design cars with a barrier between the chauffeur and the inhabitants so they can talk freely. This is how they used to do it. This arrangement is called "postilion" and the driver, instead of being in the carriage, is riding one of the horses. The downside - you needed one rider for each pair of horses, and postilion was mostly abandoned - but is still occasionally used, as seen here, by the British royal family. But if your characters don't want the coachman to overhear them...
Image source: Jorge Royan via Wikimedia Commons.
This is racing harness and is quite different! The shafts are secured to the saddle and the "collar" is a single strap around the horse's neck with a strap between its legs. There are two sets of shafts. (Again, note the ear bonnet). The cart being pulled here is called a "bicycle sulky" and weighs no more than 40 pounds, so there's no need for heavy straps on the harness. This kind of harness is only seen in modern harness racing.
And last one, I promise:
This is actually a training setup. The saddle and britches are present and the lines are tucked through the saddle, but there's no collar. This setup is used for "ground driving" a horse - which is done as part of training before putting a driving horse to a cart and also used by some people to train riding horses.