Water dripping from an exhaust pipe immediately after starting the car is a perfectly normal reaction caused by the combustion in the engine. If water still drips from the tailpipe after the engine has been running for a few minutes, it may be the sign of a serious problem.
When a car is started, combustion takes place inside the engine. During this combustion, water and carbon dioxide combine forming condensation that is often seen dripping from a tailpipe, especially on cold days. In a well-running engine, as the engine heats up, the condensation stops. This usually only happens in the morning, but can occur anytime after the car has been parked for several hours and the engine has had a chance to cool off. On warm days, or in a warmed up engine, the water released by the engine's combustion is still there in the form of invisible water vapor.
Water dripping from the tailpipe long after the engine has been running may be the sign of a more serious condition, especially when there is smoke or the smell of burning oil present. If the car has a sweet smell, like the smell of antifreeze, this is a particularly worrisome condition. Any of these signs suggest a much larger problem and the engine should be looked at by a qualified mechanic.