I love my cat.  They're often valued by us for companionship and their ability to clean up mice and such.
I used to have a cat, I do miss having a cat around the house
They are normally called house cats when kept as indoor pets, or just cats when there is no need to distinguish.
bur are cats truly domesticated like dogs?
Well under controlled breeding, they can be bred and shown as registered pedigree pets.  It's actually known as a "Cat Fancy".
Cat Fancy? is that what they call the process of breeding cats?
When they're bred and shown as pedigree pets, yes.  Cat Fancy is also a magazine dedicated to cats, owners of cats, and breeders.
when they breed pedigree cats, what characteristics are they breeding for?
Like most other mammals, cats have poorer color vision but a better sense of smell than humans.  Just a couple characteristics pedigree cats are known for.
how many registered breeds of cats are there?