Answer:
Author-title catalogue, arranged by the names of authors and the titles of books. (You could have separate author catalogue and title catalogue, but I have not seen that in practice.)
Subject catalogue, arranged by subject headings for books, such as Library of Congress Subject Headings.
Dictionary catalogue — quite common in the United States — which combined author, title and subject heading access.
Classified catalogue, arranged by the notation for a classification scheme. The one that I have seen used the Universal Decimal Classification (UDC), which is partly designed for this kind of catalogue.
