That irritating, high-pitched squealing sound from your toilet is coming from the fill valve. It might mean that the water pressure in the plumbing system is too high, but more often than not, it's due to a worn washer in the ballcock assembly. You can replace the washer, but it's just as easy and more effective to replace the entire fill valve, and it won't cost much.
A Vibrating Washer
Ballcock fill valves are the older-style ones with a ball float attached to the end of an armature. The valve that opens and closes to fill the tank is in the top of the assembly just under the point to which the armature connects, and the squealing sound comes from the washer or gasket around this valve. A noisy washer is especially likely on a metal ballcock assembly -- in part because metal ballcocks are generally older models -- but it can also occur in plastic models.
Servicing the Fill Valve
Even though the recommended repair for squealing is to replace the fill valve, you may have reasons for wanting to keep the old one. In that case, you can replace the washer.
Turn off the toilet shutoff valve and flush the toilet to drain the tank.
Unscrew the screws on the top of the valve assembly with a Phillips screwdriver, lift off the top and remove the washer and any gaskets.
Take it to a plumbing supply outlet to find replacements.
Install the new washer, gaskets and screw on the top of the valve. Turn on the water and verify that the squealing has stopped.
Replacing the Fill Valve
Replacing the the ballcock with a cup-style fill valve will stop the squealing once and for all.
Turn off the toilet shutoff valve, flush the toilet and then sponge all the water from the tank to the bowl to prevent water from falling on the floor when you remove the old valve.
Unscrew the water supply from the old fill valve. You can usually do this by hand, but if not, use adjustable pliers.
Loosen the nut under the tank that holds the old fill valve, using adjustable pliers. Remove the nut and lift out the valve.
Drop in a new valve after adjusting the height according to the instruction on the container in which it came.
Screw on the retaining nut, reattach the water supply and turn on the water.