Fortunately, this is a problem which can usually be resolved with a few tools usually found in anyone’s home and nothing more messy than getting your hands wet.

The first thing to do is to try jiggling the flush handle to see if this frees the float-ball or flapper covering the water outlet to the toilet bowl from the cistern.  This blocks the continuous water flow and returns the toilet to normal operation.

Some toilet mechanisms have a tube running vertically inside the cistern which allows the water to overflow into the bowl should the ball-float not be working properly.  Usually you can simply adjust the float-ball mechanism so the water is not allowed to rise so high against the overflow tube.  Lift the top of the cistern to gain access to the float-ball mechanism and raise the float on its arm, usually a plastic or metal spindle.  If this stops the flow of water into the toilet bowl, you simply should bend the metal spindle so the float-ball will come to a stop at a lower level.  This will stop the water from rising too high within the cistern.  Ideally, you are looking to shut the water off so it stops rising within the cistern before it reaches half-an-inch to an inch below the top of the overflow tube.

The float-ball itself may have sprung a leak so it has become filled up with water which means it does not float properly.  You can diagnose this by turning off the water supply to the cistern and then flushing the toilet until the cistern is emptied.  Unscrew the float-ball from the spindle and see if has water inside it; usually it will feel heavy if there is water inside otherwise it should feel light (it is supposed to float on water after all).  If water has seeped into the float-ball you should replace it with a new one which is easily sourced from your hardware or home improvement store.  If you have the ability, replace the entire valve with a modern flush fitting which does not rely on a float-ball at all and you’ll never have the same problem again.

Another problem is with the flapper at the bottom of the system which covers the cistern outlet and allows water to flush down into the toilet bowl.  Flappers can become distorted or more usually they have a small link chain which becomes entangled with the flapper forcing it to remain open.  You can easily replace the flapper for around $10, again from a hardware store.  This job will take you less than 15 minutes to perform, but before you attempt to replace the flapper and/or the link chain, make sure you turn off the water supply and empty the cistern by flushing the toilet.

Occasionally, replacing the flapper will not rectify the problem because the valve seal has disintegrated or lost its integrity which means the leak continues.  You can “scour” the valve seal with wire wool or you can replace the flush valve assembly, but this is a much bigger job though within the ability of most people.