The first thing is to get to understand the mechanism of your toilet. Once you understand the basics it is pretty much the same story no matter what brand of toilet you have installed in your house.
Open the tank lid and flush it a couple of times while you notice the process. You shall see that when the handle is pushed, a flapper is lifted by a chain, which results in a water fall through an opening at the bottom. The flapper later drops, closing the opening with the water level dropping.
To understand this process better you need to know that a float is connected to a valve which lets the water enter the tank.
The best way to grasp the problem is to catch it while it is happening by lifting the tank.
The first chance is that the flapper is most probably stuck, so you need to close it if you see that the tank is not filling. You can do this by your hand; all you have to do is reach in and pull.
You also need to determine if the flapper is getting caught in the chain or the chain catching on something. In most cases the chain might need changing.
In case your toilet has a ball seal, then you need to determine if the wire to lift the ball is straight and easily moveable.
Another important thing is to check if there is enough water in the tank, because if there is no or less water then this will keep the water tank running.