Purebred dogs and cats- A 'purebred' means that the genetic history of a domestic animal's ancestors has been tracked, recorded, and registered. This record is called a pedigree. If a pet is being sold as a purebred but does not have a pedigree from a recognized registry (Canadian Kennel Club, Cat Fanciers Association) then it is technically not a purebred. It is fradulent under the Canadian Pedigree Act to sell a pet as a "purebred" and not have a pedigree. It may look like a purebred, but without a pedigree you really don't know.
Purebred dogs and cats have well known physical and behavioral characteristics. This is an advantage if you desire specific physical or behavioral traits. On the other hand, purebred pet's may be predisposed to certain genetic diseases. Reputable breeders should be able to show that the pet's parents have been screened or tested for these defects as much as possible. Reputable breeders are more interested in promoting the health and welfare of the breed than generating income.
Mixed-breed pets are usually not just a generation away from purebreds but may be the result of several generations of outbreeding. They often get described as the result of a 2-purebred cross like a "Shepherd-Collie" for example, but may have the genetics of many breeds. This can reduce the chance of genetic diseases but also makes it more difficult to predict physical and behavioral characteristics. Mixed-breed cats are much more common than purebreds. You can help provide a home for shelter and rescue/abandoned pets by adopting a mixed-breed pet. Be cautious of internet rescue purchases from foreign countries. Most are legitimate, but some have the main purpose of taking your money. Get some references.
There are reliable internet kits to test the DNA of your mixed-breed dog. This helps you and your veterinarian understand your dog's health better.