A shot in the dark:
First, take the cover off the tank and make sure that the little hose from the fill valve to the flush valve is mounted above the top of the overflow riser, and isn't stuck down in the riser, creating a siphon.
Next, do this test: Take the cover off the tank. Mark the water level with a pencil. Turn off the water. Come back in an hour. Has the water level in the tank fallen? If so, then you likely need to replace the flapper (or the seal around whatever flush valve setup you have). If so, give us the manufacturer and model number stamped in the tank of the toilet, and we can help you find the right flapper or flush valve stuff. Or take a picture of the innards of the toilet tank and post it here and we can tell you what you need.
If the water level hasn't fallen, then come back here and we'll help.
PS Most likely, water is exiting the tank through a flush valve seal (flapper, tank ball, whatever) that needs replacing, and the fill valve is slowly adding water to keep up.
See toilet anatomy and such at the web site of our friends at Korky: http://korky.com/anatomy.html