Toilet shut off valve not shutting off
There are three valves in and around your toilet:


The supply valve is what connects water from your home's cold water pipes to a hose or tube that connects to your toilet tank. This is what you have pictured.
The fill valve inside your toilet tank lets water into the tank (usually sending some to the bowl also) so it is ready to flush. These are designed to fill the tank to a certain level and then shut off. The common designs for this are the traditional ballcock style with a round float attached to a lever, and the newer style (popularized by Fluidmaster) where a plastic float slides up along a shaft.
The flush valve is what dispatches water from the tank into your bowl when you flush.


The valve you've photographed is the supply valve. It does sound like your supply valve allows some water through even when tightened, which is not good because you want a supply valve that shuts off water completely for service. However, the more significant problem is your fill valve. If the supply valve leak is slow, you may just be able to shut it of as best you can, have it drip into a bucket for a few minutes, make other repairs, and reattach it for normal use.

The reason we know the problem is your fill valve is that, even when the supply is fully open, the fill valve should stop the flow of water into your toilet tank when the tank is full. If that doesn't happen, the tank will be overfilled. To prevent water spilling onto your floor in that case, the tank is designed with an emergency overflow path that just drains overflow into the bowl. Nice to not have a wet floor, but you're still wasting and paying for a lot of water.

So the question then is: why is your fill valve not stopping the flow of water into your tank? Photos may help, but the common causes are:


Incorrect adjustment. Fill valves can be adjusted to set the desired fill level, usually by turning a screw. If the level is set too high, it will never reach that level and keep filling while the overflow drains.
Obstruction. The valve control can be blocked, either by mineral buildup on the valve parts or by poor positioning next to other toilet parts.
Wear. Some fill valves include rubber gaskets/washers that erode over time, or can accumulate mineral deposits. Cleaning or replacing these parts, or sometimes simply tightening parts of the valve, can fix this. But often it's easier to simply replace the whole flush valve.


What to do next:
We want to figure out where your fill valve is leaking, so let's try to control its operation so we can observe it.


Take the lid off your tank.
Open the supply valve.
Standing over the toilet, gently lift the fill valve's float mechanism to the point where it should be shut off. Hold it there - the fill valve should stay closed.
Hold the flush valve to drain as much water as possible from the tank.
Identify where water is exiting the fill valve. This is your leak.


Depending on your type of fill valve, you may be able to repair the leak or it may be more practical to simply replace the valve. Post pictures and info on the leak if you want more advice. Good luck!