Ships must follow an extraordinary number of rules and regulations in place to protect passengers' and crewmembers' safety while onboard. The Coast Guard conducts rigorous, quarterly inspections of all ships that operate from U.S. ports, looking to make sure they comply with emergency response requirements. Ships also operate under international rules, known as Safety of Life at Sea (SOLAS). The rules regulate everything from fire safety to navigation and maritime security. The rules also require that before the ship sails, everyone must participate in a safety drill that includes instructions on locating and putting on life jackets and finding your assigned lifeboat.
The biggest safety concerns for cruise ships are fire and running aground/capsizing. The past few years have seen some high profile and tragic incidents, including onboard fires on Oceania Insignia, Royal Caribbean's Grandeur of the Seas and Carnival Triumph and the capsizing of Costa Concordia, but these occurrences are actually quite rare, given the millions of people who cruise each year. You may be more at risk driving to the airport or boarding a plane to get to your homeport than you are once at sea.
In the aftermath of the Concordia tragedy, the Cruise Lines International Association adopted a cruise industry passenger bill of rights. The bill guarantees passengers rights in 10 areas including safety, comfort and care. Rights include "a full refund for a trip that is canceled due to mechanical failures or a partial refund for voyages that are terminated early due to these failures" and "timely information updates as to any adjustments in the itinerary of the ship in the event of a mechanical failure or emergency, as well as timely updates of the status of efforts to address mechanical failures."
Also note that cruise ships are like mini-cities, and you should take the same general travel precautions you would on land. Keep any valuables in your cabin's safe (or leave them at home), and don't open your cabin door without verifying who's there. Parents, give children strict rules about when they can and cannot roam the ship without adult supervision -- especially near the swimming pool.