Also spelled moggie and commonly referred to as a domestic shorthair or long hair, a moggy is a cat of unknown origin/mixed breed. They are the most common type of domesticated cats and come in all shapes, sizes, and colours. Moggies come in both long and short hair. Big, small, fat, thin, talkative, quiet, friendly, shy, blue eyes, green eyes, yellow eyes, gold eyes, odd eyes.
For the most part, moggies are a healthy bunch due to their wide gene pool. They are easy to obtain and cheaper to purchase than purebred cats but sadly, they are harder to find homes for.
Felis silvestris lybica
All domesticated cats are descended from their wild ancestor African Wildcat (Felis silvestris lybica). Humans domesticated cats when we began farming. It is now believed that cats and humans began their relationship around the Fertile Crescent 10,000 years ago. 
How the separate breeds came about is pure speculation (on my part), but I would imagine that once man and cat began to live in close proximity, and as we spread throughout the world, cats came along for the ride and different mutations occurred. It would stand to reason that a cat living in Russia or Scandinavia would need to have a long, thick coat, a cat living in southern Asia wouldn’t. Cats changed and adapted. Many of the breeds we know and love today are the result of spontaneous mutations. The Devon and Cornish Rex, Scottish Fold, Sphynx, while other breeds are man-made such as the Havana, the Australian Mist, the Ocicat. It, therefore, seems likely that mutations have occurred in the previous 10,000 years since man domesticated cat (or did the cat domesticate man?) resulting in the many different cat types and breeds we see today.
If we stopped selectively breeding cats, they would eventually revert back to their wild type, which is the tabby cat.
Sometimes moggies are used in breeding programmes to increase the genetic pool or introduce a specific trait. Allowable outcrosses include American Curl, Devon Rex, La Perm, Munchkin. This list may vary from country to country and breed to breed.
Why the name moggy?
Some say it is derived from the name mongrel, but it is generally accepted that the name originates from Margaret or Maggie. This was a term used to describe a disheveled old woman.
Can I show my moggy?
Yes, most cat shows have a section for moggies known as household pets. If your cat has the looks and temperament, why not give it a try? They are not just for the purebreds.