Barring a crack in the porcelain -- which means you need a new toilet -- a toilet tank won't fill for one of two reasons. The first is that the fill valve isn't working, and that could be related to the water pressure in your house. The second reason is a leaky flush valve, which isn't difficult to diagnose. A simple adjustment or replacement of an inexpensive part usually is all that's necessary.
Low Water Pressure
When you flush the toilet, you should hear a rush of water coming into the tank from the fill valve. Failure to hear that sound may mean the toilet shut-off valve isn't open all the way -- someone may have closed it to clean the toilet or make a repair. The valve is under the tank. Turn the handle all the way counterclockwise. If the flow doesn't improve, check the water pressure at the sink faucet. If it's low and your house is serviced by a well, the pump or pressure tank could be malfunctioning.
Adjust the Fill Valve
All toilet fill valves have a float that controls the water level. Turn a screw with your fingers or a screwdriver, unhook a tab or if it's a ballcock, rotate the ball on the end of an armature to adjust it. If the float is adjusted too high, the fill valve shuts off before the tank fills. Turn the screw counterclockwise to raise the float; if it's a ball, rotate the ball counterclockwise with respect to the armature. If it has a tab, unhook the tab and lift the float.
Replace the Fill Valve
If the water pressure is adequate, the float is set at the proper level and the tank still doesn't fill, the fill valve itself may be encrusted with scale or be broken. If so, install a new fill valve. Turn off the water and disconnect the supply hose, then unscrew the locknut under the tank and lift out the old valve. Before installing the new valve, follow the manufacturer's instructions for adjusting its length to suit your tank. Use adjustable pliers to loosen and tighten the locknut that secures the valve to the tank.
Replace the Flapper
Your tank may be filling normally, but water may be leaking through the flush valve into the bowl. If the leak is fast enough, the fill valve won't stop running, and if the leak is a slow one, the fill valve will cycle on and off. The problem is the seal formed by the flapper isn't secure. Flappers are the easiest of toilet parts to replace -- just disconnect the chain, pull or slide the old one off and reverse the procedure to install the new one. Make sure the new flapper is the same style as the old one.
Replace the Flush Valve
If the flush valve is old, it may be cracked or it may not be able to form a good seal; in that case, replace the valve with a new one. This involves disconnecting the tank from the bowl so you can unscrew the old flush valve and install the new one. When you replace the tank, it's a good idea to use fresh bolts and a fresh rubber washer to seal the aperture between the tank and bowl.