The float ball lifts up on the arm to close off the flow of water into the tank.
What happens if you lift up on the arm directly? Does it stop the water completely?
Sometimes the rod to the float can be bent so that it allows the water level to go higher than it should.
If the tank fills too high the excess water will flow over the top of the tube and drain so that it does not overflow the tank, do you see that happening?
If it never gets that high then it sounds like it is not overfilling so the flapper at the bottom is probably letting water leak out enough that the toilet has to keep topping itself off to stay full.
If lifting the float arm does not work to shut off the valve then the valve probably needs replacing. If it does shut off the flow of water then perhaps the float arm needs more adjusting than the screw allowed for, perhaps due to a bent rod to the float and it might be possible to bend the arm back down to get the rest of the adjustment.
If the flow of water into the tank stops but the level slowly drops or you can even hear trickling of water then the flapper may need to be replaced.
One of the resident experts can tell you more but I have had to look at these problems so many times in my own homes that the above will most likely cover the problem even if I do not have the terminology right myself.
