You've got one of two different problems. Either the water is not getting into the tank, or it's leaking out into the bowl (or both). To check if it's leaking, you can put some food coloring in the bottom of the tank, and see if leaks into the bowl (that's the high tech way). The low tech way is to listen - if you hear water running, and the tank is not filling up, the water must be going someplace, almost certainly into the bowl (unless it's leaking onto the floor

). Take a look at how the big rubber flap fits on that big hole in the bottom center of the tank. Maybe some debris got in there so it doesn't seat correctly.
The other problem is caused by some sort of faulty cutoff. When the toilet is flushed, water leaves the tank and enters the bowl. There is a big ol' switch almost always controled by a big ol plastic ball for a floater. When it gets high enough, the toilet "thinks" the tank is full, and cuts off the water. Check and make sure that all works. There's plenty of ways to adjust it, and it's pretty basic.
The other cutoff problem (that is more common actually) is that the cutoff is *not* working, the tank keeps filling, and the overfill flows into the bowl. Usually that is caused by a leaky float (it fills with water, and won't shut the intake off). But I don't think that is your problem - you should still be able to flush the toilet in that case.