By the very definition of the word, emergencies are things that need to be acted upon as quickly as possible and plumbing emergencies are no different. Whilst dripping taps or shower heads may not be things that need to be fixed immediately some plumbing issues have to ability to not only damage your house and belongings but they can negatively affect your health as well.
If your toilet gets clogged and the water is no longer draining out of the toilet when you flush, turn off the water supply to the toilet. Attempt to unclog the toilet with a plunger, if this is unsuccessful you should call a plumber.
If a drain stops allowing water to freely flow down it, turn off the tap associated with the drain, try to remove the blockage with a rubber plunger. The plunger should be large enough to completely cover the opening of the drain and the water should come up the edge of the rubber cup. If the plunger has not worked then you should call a plumber, DO NOT attempt to use a drain snake or any and chemical drain cleaners as these may cause more damage to your pipes.
If a tap continues to run after you have turned it off, switch off the water supply to the tap. If you are experienced in diagnosing leaks and replacing relevant seals etc. then do so, otherwise call a plumber.
Pipes do occasionally leak, burst and break, and when they do they need to be fixed as soon as possible. First you should shut off the main water supply, this will stop the water flow to the house and stop and further water from leaking out of the pipe. Next turn of the power / gas supply to your water heater so it doesn't over heat. Call a plumber, one who specialises in non-destructive leak detection may be necessary if the leak is in a wall or under concrete.
In the event of a major blockage or pipe burst flooding can occur, you must shut off your electricity supply and water supply to the house for safety purposes. Call a plumber immediately, you may also need to call a handy man or builder to help pump the water out of your house or repair any water damage that has occurred.
If you know what to do in an emergency situation before it happens and have a store of supplies that you will need in an emergency the experience becomes a lot less stressful. It is important to note that while other plumbing issues may not be classed as an emergency, they are still important to get fixed by a licensed plumber; many issues can grow to become much larger, more dangerous problems quickly.