Moggy cats are random breed, cross-bred non-pedigree cats and the opposite to moggy is the purebred, selectively breed, sometimes inbred pedigree cat. These are sold by breeders while moggies are given away or they come up to you in the street to be adopted or they are adopted from shelters at a modest fee.
Before around the mid-1800s all domestic cats were moggies per modern standards. Then the idea of selectively breeding cats came about and cat breeds were created and became popular. The oldest cat breeds are Persians, Siamese, Maine Coons, British Shorthairs and Turkish Angoras. In general these date back to the late 1800s. However, the truest and purest Turkish Angoras (true Turkish Angoras) are in Turkey on the streets and technically moggies.
As is the case for many cat breeds they started out as moggies and then refined by selective breeding. The Maine Coon was a moggy barn cat in Maine in the 1850s and then quite soon became a superb looking purebred cat in the early 1900s. It is now probably the most popular purebred cat. Sometimes breeders overdo the selective breeding and end up with a monster of a cat such as the rat-faced modern Siamese or the flat-faced modern Persian. It can go wrong but the cat associations who are responsible for these overbreeding never admit it. Never.
The prized purebred cats suffer from inbreeding which shortens their lives while the badly bred moggy enjoys ‘hybrid vigour’ which improves physical stamina and creates longer lives. The longest living domestic cats are moggies in loving homes. Often moggies live on the street as community cats which shortens their lives.