Time to get this fixed, a toilet running like that can create a major hit on your water/sewer bill, or mess up a septic system fast.
ANyway, unless it is a pressure assist toilet, they almost all have a rubber flapper that is lifted when you push the lever down to flush the thing. Sounds like this is not seating properly and letting the water leak out faster than it can be filled up. It could be other things, but that is the first place to look. Sometimes, the chain that connects it to the lever gets kinked, and the thing is just hanging there a little above the seat (the place where the flap sits). These flapper valves often get soft and spongy, and don't make good seals anymore. If it is old, it wouldn't hurt to replace it, and it may be the source of your problem. While you are looking, check the seat - it should be smooth all the way around, or the rubber flapper valve won't be able to seal when it falls after the flush. Once in awhile, one of the arms of the flapper valve gets yanked off of one of the nubs that hold it to the overflow tube. If so, just replace it over the nub (unless it split - then replace the flapper valve).
The last thing I can think of, is that the overflow pipe might (and this is rare) have a crack or hole in it, and thus letting all of th ewater from above that point flow down into the toilet, and thus, the tank can't refill and shut off.
If you had said that it didn't shut off at all, and the tank had reached the top and was running down the top of the overflow, then it would be the filler valve assembly - the thing is not either adjusted right, or the valve is leaking, thus even though it is being told to turn off, it keeps running - like a leaky faucet.
The most common thing is for the flapper valve to be worn out and needing replacement.
While many will work, if it is a low-flow toilet, it is best to replace it with the manufacurer's brand, or one that is an exact fit. Otherwise, your low-flow toilet may flush lots more water (or not) depending on the flapper valve. But, if you can't find one of those right away, any flapper valve that fits (and it it fairly common in size between most brands) will save you far more water than worrying about the exact brand if you can't find it right away.