During a massive thunderstorm, the power goes out. In the darkness, you hear the toilet tank start running as if it has been flushed, even though you're home alone. Try not to panic -- it's just a phantom flush. A poorly seated or completely worn flapper can let enough water into the toilet bowl to lower the float until the fill valve in the tank opens, refilling the tank over and over again throughout the day. The only solution for a phantom flush caused by a worn flapper is a new flapper.
Continuous filling, like the phantom flush, is often caused by a worn out flapper. You may hear water running constantly or see the bowl fill slowly (it may also self-flush if the leak is really bad). Jiggling the handle can sometimes make this condition stop temporarily. Remove the cover from the tank and see if the flapper seats completely after each flush. If so, you need a new flapper. If not, the flapper linkage chain may be too short or bound -- it should hang just barely slack when the flapper is sealed.
The opposite of the continuous flush is the incomplete flush. It may be the most annoying of the flushes. When you flush the toilet, the flapper is supposed to open and allow all the water in the tank to rush into the bowl at once. In the case of the incomplete flush, the flapper is closing too quickly and not allowing all the water out of the tank. This is often caused by a waterlogged flapper -- because of the added weight of the water between the layers of rubber, the flapper slams shut before the tank's water has drained. Check the linkage chain for potential problems, but this is usually a good reason to replace the flapper.
Under rare circumstances, a malfunctioning flapper may result in absolutely nothing happening. When you flush the toilet, the flapper fails to open and neither tank nor bowl empties. Check the linkage first, it may have come loose from the flapper or handle. Your flapper could be also be stuck due to years of unusual wear or particularly hard water deposits. Replace the flapper if you have any doubt.
It is easy to check for a leaking flapper. Special dye tablets and products are available to color the water in the tank, but dark-colored food dyes work just as well. Put a few drops of food coloring into the tank and let the colored water stand for 10 to 20 minutes. Check the bowl for colored water seepage. Any color in the bowl indicates that the flapper needs replacement. Because flappers and food dyes are both inexpensive, it is worth the effort to perform this check once every few months -- the savings in lost water can be impressive.