Check water supply
Make sure the water supply is turned on in the house and at the toilet.
The handle is loose, stuck or damaged
If the tank handle is loose it can be tightened by using an adjustable wrench to turn the locknut on the inside of the tank counterclockwise.
If the nut cannot be turned you might try applying a lubricating oil and allow it to sit for several minutes before attempting to tighten the nut again. If that does not work you may need to remove the nut by cutting through the handle shaft with a hacksaw. You will then need to replace the handle and trip lever, which is attached. Simply remove the trip lever from the chain and slip it through the toilet tank hole before replacing it with a new handle assembly.
Lift chain or wire is out of place or damaged
If the lift chain in the tank it too short it may be allowing water to seep through the flapper valve. If there is excess chain you can use pliers to open the links and reposition the chain. If the chain is too short you will need to replace it. Using a paper clip or other other wire to lengthen the chain will cause it to corrode.
If your tank assembly has a lift wire you can loosen the screw attached to the guide arm and move the guide arm up or down to adjust the length.
Ball cock is damaged
The ball cock opens and closes as the float ball falls and rises with the water level in the tank. This action controls the flow of water into the tank. With a plunger-style ball cock the float arm pushes the valve plunger and washer to stop the water flow. With a diaphragm ball cock the plunger pushes against a rubber diaphragm.
Before working on the ball cock turn of the water and flush the toilet to empty the bowl. Remove the screws surrounding the ball cock assembly to gain access to the washers or diaphragm. You may need to slide the float arm out of the way to gain access to the assembly.
Remove the washers or diaphragm from the valve plunger with a small screwdriver. Use vinegar and a small brush to clean sediment from inside the ball cock and washer or diaphragm. Once cleaned reassemble the parts and the ball cock. If any of the pieces appears damaged or if the water continues to run, replace the ball cock.