Per wikipedia, Carnival's corporate family carries 42% of all cruise passengers.
Royal Caribbean's corporate family carries 23.6%, so those two companies carry ~2/3 of all cruise traffic. As far as individual cruise lines go, it looks like RC narrowly edges out Carnival for most passengers (19.1% to 18.2%).
A cruise line is a company that operates cruise ships and markets cruises to the public. Cruise lines are distinct from passenger lines which are primarily concerned with transportation of their passengers. Cruise lines have a dual character; they are partly in the transportation business, and...