Quarter Horses were bred for use on ranches, primarily to assist in herding cattle. Cutting cattle from the herd and dealing with them requires short bursts of speed. Humans, of course, race everything.
Because the early "stock horses" were built to sprint over very short distances, they were raced over short distances - which became standardized to a quarter of a mile.
So, a Quarter Horse is a horse that runs its fastest over a distance of about a quarter of a mile. Thoroughbreds, in contrast, race over distances ranging from four furlongs (half a mile) to a couple of miles.
The Quarter Horse remains popular because of its versatility.
The horse in the foreground here is PWF's Gooseberry, a "foundation-bred" Quarter Horse.