What Does "Registered Cat" Mean?
In its narrowest sense, a registered cat is a cat which has had its birth registered with one or more of the associations which register cats. And that's it. A registered cat is not the same as a show cat, nor is registration a guarantee of the cat's health. What it is a guarantee of the cat's lineage.
On the other hand, technically speaking, if a cat is not registered, it is not a "purebred". That is because you cannot be sure about the breed of its parents without registration. So, if a cat is called a pure-bred, but is not registered, and it looks like a Burmese, for example, what you have is a cat which appears to be a Burmese. You cannot be certain that it is one.
That mean when you see a "pure-bred" cat for sale at a pet store, don't assume that it is a registered cat. And, if you see a "registered" cat at a pet store, please don't assume it is also a "show" cat. As you will soon learn, a show cat is also something different.
If you go to a pet shop, and ask about registered cats, a reputable pet store owner may tell you something like what I was told when I was preparing this piece. The owner said "A registered cat means I can sell it for more."
So, from your point of view, buying a registered Persian means you have bought a pure-bred Persian. It does not necessarily mean that the parents were show cats. It does mean, however, that the parents are also registered.