Toilets and water tanks are fed by water via a valve which automatically stops when they are full.
In the event of the valve becoming faulty you should find an overflow pipe which diverts the water outside. You may see this dripping as the fault begins then really starts to flow as the valve starts to fail badly. Even if you just see it dripping that is the time to take action. When buying a replacement ensure it is suitable for the pressure of water you have coming in.

In some modern toilet cisterns the siphon diverts water into the pan when the valve fails so you might see a constant trickle going down the pan (and wasting water).

Toilet Cistern
Lift the cistern lid and you should see the water level much higher than normal. You should see the water flowing out the overflow pipe. (or with a modern siphon into the pan) You will need to change the valve.

Turn off the water. And flush the toilet. There are several types of valve. The traditional type is a round ball cock float on a long arm to the shut off valve.

Using a footprint wrench or adjustable spanner unscrew the bottom of the valve from the water intake pipe. Then unscrew it from the cistern.
Replace it with a new valve.
Then turn the water back on and adjust the valve so that the water level comes back to it's original height(you'll see some staining on the side of the cistern)

The adjustment is either on the float or the body of the valve depending on which type of valve you choose. e.g. on a Torbeck valve you can screw the small float up and down. With a plastic ball valve there is a small nut and bolt assembly which is adjustable to give the correct level.

If you do have an old ball cock type you may like to replace it with a more modern type which tend to fill more quickly and quietly.(like a Torbeck or Fluidmaster valve) It has a tiny float compared to a ball cock type.

Water tank
Changing the valve is very similar to the toilet cistern. They are simpler to change other than you may be pressed for space if it is in the loft and often the retaining nuts can be difficult to remove if they are seized up. (try spraying the retaining nut with WD40)
Valves on water tanks will all be side entry which makes it easier as you won't have to drain the tank. Turn off the water first.

If you change the valve like for like (they are often a brass Portsmouth valves which are very cheap and reliable) then the water level should come back pretty much to the same level as the original without much adjustment. (which is handy if your tank is awkwardly placed in the loft.)