A purebred dog or cat is one with a known heritage where all the ancestors were dogs or cats of the same breed. This known heritage can usually be tracked back through a pedigree on registered purebreds but it may only be word of mouth on the breeding background, plus the physical appearance and behavior of the particular animal that is being considered.
Some people mistake color differences as different breeds and some mistake cross breeds as a breed.
For example I recently saw an ad for a ‘mix’ between a black and yellow Labrador stating the pup looked just like a black Lab. And well it should look like a black Labrador Retriever as black, yellow, and chocolate are all simply colors of the same breed and the pup was not a mix at all even though its parents were different colors.
Another color difference often mistakenly described as a breed is seen in pit bull terrier type dogs. You might see ‘blue nose’, ‘red nose’ advertised as if they were different breeds from the other colors of pit bulls but they are not different in any way other than color. You may also see famous kennel names attached to certain dogs but this does not make them a different breed just the product of one kennel’s breeding program for the same breed. For example a Razor pit bull is not a new or different breed.
I also frequently see mixes advertised as purebreds. But the fact that they are mixed means they are not purebred anything.
A Siamese is a purebred but an ‘all black Siamese’ doesn’t exist and is most likely a cross between a purebred Siamese and some other type of cat.
A Pekingese is a purebred dog as is a Poodle but a Peke-a-poo is a mix not a new breed. Even if you breed two Peke-a-poos together you do not have a breed. It takes more than one generation to create a breed and its not as easy as it sounds to cross dogs and get consistent results.
The same holds true for a Labradoodle or GoldenDoodle these are simply mixes between purebred Poodles and Labradors or Golden Retrievers. While there was an attempt in Australia to create a new breed by these sorts of crossings, through many generations of breeding and careful selection of offspring for specific traits, it did not work and the breeding program was discontinued because they had no luck creating what they hoped for in a new breed.
It takes 6 generations of the dogs breeding true, to a type defined in a written standard, to be able to consider a dog a purebred. By true to type I mean the offspring mature as adults to look and act like the parents in most ways. Cat breeders have similar rules to follow.
You can see AKC standards for dogs here by clicking on the breed name
http://www.akc.org/breeds/complete_breed_list.cfm
And CFA standards for cats here
http://www.cfainc.org/breeds.html
While you can create a new type if you have a mutation to work with, for example American Hairless Terriers and curly coated cats, or a specific type in a standard that you can reproduce consistently, simply crossing already established breeds doesn’t do it.
When you are seeking a pet you can often weed out breeders and even rescues that are not the most informed, based on how they advertise. If they claim a mix is a breed, or that a purebred is a mix, or special because of its color, then you have a fair idea that this could be someone who isn’t really an expert on the animals they are trying to sell.
