Most big cruise ships have security guards, but there is nothing like an organized police force on a ship. The staff captain is usually charged with "maintaining order" onboard, and would discipline a crewmember (especially) or could "arrest" a guest if they caused trouble.
In all cases the captain is ultimately responsible for the "law" at sea. By international maritime law he has the final word on what happens on a ship, within reason of course. He is not allowed to maltreat a crewmember in the name of corporeal punishment, for example, but he can definitely decide who gets held under arrest, or let go. He can put people off the ship for any reason.
The security guards are basically like mall cops - they are there to make sure no one gets out of line, but they are basically civilians like everyone else. It is their job to get involved when there is trouble, and they would probably be given the consideration of a legal peace officer if they had to use brute force to subdue a dangerous person. But if they made a mistake and accosted someone who turned out to be innocent they could also be sued pretty easily.
They have to answer to the superior officers onboard - they cannot arrest the captain, for example.