A pedigreed cat is a cat with “the papers”—specifically, registration papers listing the cat’s purebred parents, grandparents and great-grandparents (and even further, if that information is available).
In other words, it is a family tree that ensures that the cat is not a mixed breed (not that the cat himself gives a hoot). The papers will also list the cat’s ancestors’ titles—awards won in cat shows, that is.