This may be caused by a leaking flush valve, allowing water to pass through the tank into the bowl. If this is the case, you will hear the toilet start filling all on it's own from time to time.This is because as the water leaks in to the bowl, the tank level drops and the toilet thinks it has been flushed, so it re-fills.
If the flush valve leaks the flapper must be replaced and the seat it sits on cleaned thoroughly.
If the toilet always sounds like it is filling a little (a constant hiss of running water) then the problem likely lies in the fill valve mechanism (ballcock valve). Lift tank lid.
If the fill valve is leaking you will see water coming up over the overflow tube and entering the bowl. Lift up lightly on float rod and ball. If water stops running when you lift only slightly you are in luck! You can adjust the water level in the tank by holding tightly to the rod near the ballcock and bending the rod downward slightly. Be careful not to put too much stress on the connection of the rod to the ballcock valve. If the hissing of the incoming water does not stop when you lift up slightly on the float then you must continue to the next step.
Shut off water to toilet, unscew the float ball and shake it. If you can hear water inside the ball, throw it away. They are cheap and impossible to dry out once partially filled with water. The other cause of constantly running toilets, and probably the most common is malfunction of the fill valve. They contain a small diphragm (called the ballcock diaphragm or fill-valve diaphragm) that is controlled by the lever action of the float valve. This diaphragm is what shuts off the water and as they get old, or some debris gets caught in them, they leak. These diaphragms are very cheap and easy to find at most hardware stores. See our Exploded View of a Toilet