A horse does not need to be registered to be bred. However, if a horse isn't registered, it may be more difficult to register its offspring in a breed association.
Registering a horse to a breed registry is similar to having American Kennel Club papers for a dog; it's simply a way to say the horse belongs to a breed association and is purebred. If a horse's parents weren't registered, the association may require a DNA test to prove its breed.
Membership in a breed association is not mandatory, however associations like the American Quarter Horse Association often include perks and special events. For example, the AQHA only allows registered horses to compete in AQHA shows and races and gives members discounts at a variety of businesses such as John Deere and Ford.