These terms are often confused or used inappropriately. Here we provide some information and links to help clarify the situation.
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Purebred: According to the American Kennel Club, for a dog to be purebred means "the sire and dam of a dog are members of a recognized breed and the ancestry of a dog consists of the same breed over many generations." Wikipedia typically defines a purebred dog as “a dog of a modern dog breed with a documented pedigree in a stud book and may be registered with a breed club that may also be part of a national kennel club.” Often used interchangeably with “pedigreed”.
Pedigree: A pedigree dog is a purebred dog that is eligible for registration with a recognized kennel club or society that maintains a register for dogs of that description. A pedigree itself is a record of a dog’s family tree – its sire, dam, grand-sires, grand-dams, etc.
Crossbreed: Also called designer dogs, mixed-breed, and random-bred dogs, crossbreed dogs are the result of breeding different purebreds or other mixed-breed dogs together.
Designer: Designer dogs are, essentially, simply crossbreeds – though “designer” generally has a more negative connotation than “crossbreed”. According to Wikipedia, the key identifier of a designer dog is that the “breed” is a word made up of syllables from the breed names of the two purebred parents, such as Schnoodle (Schnauzer and poodle cross).
Breed: According to Wikipedia, breeds are groups of closely related and visibly similar domestic dogs, which are all of the subspecies Canis lupus familiaris, having characteristic traits that are selected and maintained by humans, bred out of a known foundation stock.
Two very different definitions exist:
1. Pedigreed dogs that are bred on purpose.
2. Dogs of any kind bred with a specific purpose in mind, be it guide/assistance dogs (http://warriorcanineconnection.org/our-service-dogs/purpose-bred/), working sheepdogs, or even dogs destined for use in research.