What should I do if water is dripping from the ceiling inside my house?

The first task is to figure out if the leak is coming from inside of your pipes or from external moisture like rain.  If the leak only happens when it is raining, then shut off the main water supply and see if the leak still happens.  If yes, then it is likely from rain.  If it has been sunny and dry for a week and you are still getting the leak, shut the water off and see if the leak stops.  If it does, then that is likely from an internal leak.  If it is in fact an internal leak, then call a reputable local plumber to have them test your pipes for pressure loss.  If it is an external leak, then wait for sunny weather.  After about a week of dry sunny weather, get a garden hose and pick a feature of your roof or external walls and spray that feature (vent pipe, window, roof juncture, etc) with a direct stream of high volume and low velocity water from the garden hose.  Wait for a few hours, and see if the leak appears.  After four hours, if no leak has appeared, then move on to the next feature.  If hosing down a specific feature does in fact cause dripping within that four hour window, then either call a roof repair company or go to your local hardware store and talk to an expert about ways to seal cracks or gaps in siding or roofing material.