Updated January 8, 2020
One could argue that "cruise ship time" is a whimsical way of referring to freedom from time pressures while on a cruise. You may, however, find that the term specifically refers to the hour of the day on the ship, as opposed to the hour of the day in any specific port of call.
Cruising across time zones can be tricky for cruisers. You know what time it was in port where you originally boarded the ship, and in some cases, that may be the time both you and the ship will operate on throughout your cruise.
The Caribbean would get particularly confusing if ships changed to local time at each port of call. Even though time zones are fairly straightforward, the waters get a bit muddied when it comes to daylight saving time. Most Caribbean cruise ports do not set clocks forward in the spring and back in the fall, but a few do. Because of that, some Caribbean itineraries would cause passengers to reset their watches back and forth several times during a short cruise. Many ships in the Caribbean simply advise passengers to stay on cruise ship time when going ashore. That is not usually the case elsewhere around the globe.
Transatlantic and transpacific cruises change the ship's time at night in conjunction with the crossing of each time zone -- so usually one hour per day. European cruises typically match ship's time to the port time, unless there are extenuating circumstances. These changes will also be done at night.
The good news is that cruise ships are adept at keeping you in sync. In addition to posting the current time conspicuously at the gangway when you depart the ship in a port of call, announcements are made before time changes, notices are posted in the daily bulletin and perhaps even a gentle reminder note will be placed on your bed -- all designed to give you the freedom to enjoy as much cruise ship time as you can.