Like everything else with horse terminology, there are regional definitions. What I'm calling a patterned horse is one that has patches of dark or light color and/or spots on its coat. Humans have bred spotted or patched coats into all of our domestic animals, and horses are no exception.
Spotted or patterned horses are split into two groups:
1. Horses with large patches of color on a white coat (or vice versa).
2. Horses with either small spots of color on a white coat or vice versa, or a frosted blanket over the hindquarters.
The first group are called "pintos" in the United States. You may also hear them called "paints" - but that's a specific breed and the term should really not be used for horses not registered with the American Paint Horse Association. In England they are called piebald if their color is black, skewbald if it is anything else.
A pinto draft cross.
The second group are referred to as "spotted" horses in England. Confusingly, though, most Americans use "spotted" for any horse with markings across its body. In America, they are usually called appaloosas, whether or not they are of the Appaloosa breed.
An Appaloosa mare with a "spotted blanket".
A horse can also fall into both groups, which is generally called a "pintaloosa."
In the United States some more precise terms are used for different kinds of pinto.
Tobiano horses have markings somewhat similar to tuxedo or spotted cats.
A tobiano American Paint.
Frame overo horses have markings that are "framed" by the edge of the body. This is sometimes just called overo, but overo is also used for any pinto horse that is not tobiano. The mnemonic I was taught was that an overo horse has no white "over" the back - white on overo horses never crosses the line of the spine on the back.
Image source: www.horsevet.co.uk
Sabino horses have white that starts on the belly and flanks. In most cases it stays there, but some sabino horses only have dark coloring on the top of their heads (medicine hat) and a few are pure white. Others have white hair through their body. Confusingly, sabino is called roan (along with true roan) in the UK.
Sabino is classic for the Clydesdale breed as this horse demonstrates. Image source: USDA.
Splashed white horses have white across the front part of the belly and the chest.
Horses with more than one kind of pinto are sometimes called tovero. The black and white pinto above is either tobiano or tovero - it's sometimes hard to tell.
There are also specific terms for Appaloosa horses.
Leopard - spots evenly over the body.
A black leopard Noriker mare with her spotted blanket foal. Source: 4028mdk09 via Wikimedia Commons
Snowflake - a reversed leopard, with the base color dark and the spots white. Extremely rare - I wish I could provide a picture but I can't find any that are legal for use.
Blanket - a white blanket over the horse's back.
This Appaloosa has a small blanket. Source: Brittany Hogan via Wikimedia Commons.
Spotted blanket - the horse has a blanket, then has spots in it.
This is a classic foundation Appaloosa with a very typical spotted blanket. Image source: RL65 via Wikimedia Commons.
Fewspot - the horse is almost white with a few spots in it. (This usually comes with two copies of the gene that causes Appaloosa spotting).
Snowcap - a wider blanket, also associated with two copies of the gene.
The mare shown above combines fewspot and snowcap.
Finally, some horses have birdcatcher spots, which shouldn't be confused with appaloosa spots. They are white spots that appear with age and become larger over time. Note that this terminology only dates to the nineteenth century, after a horse called Birdcatcher.
This horse has particularly large birdcatcher spots - they're generally smaller. Image source: Kersti Nebelsiek via Wikimedia Commons.
(All unlabeled images taken by me).