Inside the Tank
Isolating what's wrong with your toilet requires understanding what happens when you flush. Take the lid off the tank behind the toilet, then press the handle to flush the toilet.
You'll see that the handle is connected by a chain or rod to a rubber stopper that covers the opening to the toilet bowl. When you press the handle, the stopper comes up, and the water in the tank rushes into the bowl.
Once the water level falls, the stopper drops back into place, and the tank begins filling with water again. The water comes in through a supply line that's connected to the toilet's fill valve.
The fill valve has a float attached to it. As the toilet fills, the float rises.
When the tank is full, the float should shut off the valve.
If the stopper at the bottom of the toilet tank isn't creating a tight seal, water may be leaking through into the bowl. If a lot of water is getting past the stopper, the water level in the tank will never rise high enough for the float to shut off the fill valve, so the toilet will just keep filling and filling, never stopping.
Check to make sure the stopper is properly seated on the opening to the bowl. Sometimes simply adjusting the stopper will fix the problem, but if the stopper is worn or cracked, you may need to replace it.
You can find replacements -- with instructions included -- at hardware stores or home centers. If you can't tell whether the stopper is leaking, flush the toilet, let the stopper drop back into place, and then put a few drops of food coloring in the tank, near the stopper.
If the color shows up in the bowl, then the problem is your stopper.
If the stopper isn't the problem, then check the float on the fill valve. If the float isn't set correctly, it might not rise high enough to shut off the water, so the tank keeps filling and filling.
With the tank lid off, flush the toilet again and watch as the water rises. The typical toilet tank has an "overflow tube," a vertical tube with an opening set a little higher than where the water level should be in a full tank.
If the water rises high enough that it begins spilling into this tube, you know that your float isn't set properly. Water that flows into the overflow tube goes directly into the bowl, bypassing the stopper.
This keeps the water from rising to the top of the tank and spilling onto the floor. But it also means that the fill valve can just keep filling without ever stopping.
Adjusting a Float
In most cases, the float will either be an air-filled ball connected by a rod to the fill valve or a doughnut-shaped piece that circles the fill valve like a collar. For ball floats, check to see if the rod has threading -- like you'd see on a screw or bolt -- where it connects to the valve.
If it does, rotate the ball assembly counterclockwise a little bit at a time, as if you're "uncrewing" the rod. This will lengthen the rod, which will set the float lower.
If it doesn't have threads, you may have to actually bend the rod downward to set the float lower. A collar-style float, meanwhile, is connected to a rod with an adjustable clip.
The rod shuts off the valve, so the higher you set the clip on the rod, the higher the water level required to shut off the valve. Too high, and water goes into the overflow tube.
Set the clip lower on the rod.