Grade is a U.S. specific term to refer to a horse of unknown breeding, of no specific breeding, or otherwise not able to be registered as a purebred (or part bred) animal.
If a horse's breed is given with "grade" before it, then either the breeding is known but the proof of it has been lost or the horse is reasonably believed by educated people to be of that breed in whole or in part by virtue of its type and build. So, a "grade Quarter Horse" is a horse that looks and goes like a Quarter Horse, but its history is unknown. (In contrast, an "unregistered Quarter Horse" would be a horse bred from two Quarter Horse parents but never, for some reason, registered).
Grade is sometimes used in casual conversation the way "mutt" is with dogs, however, some horsemen make the case that you can't ride a set of papers and a well put together grade is worth no less than a purebred.