There are several reasons why a dryer may squeal and squeak when it is running. One of the most common causes is the rubber belt that turns the circular drum of the dryer has lost some of its adhesion and is slipping as it turns, making that annoying squeal. All belts must be replaced eventually, but you can extend the life of one significantly with an easy fix called "belt dressing.'' It comes in a spray can and makes the underside of the belt sticky again so that it stops slipping. Belt dressing is available in most hardware stores.
Unplug your dryer, pull it out into an open space, and open the exterior frame to get at the circular drum inside. Different makes of dryer are accessed in different ways, from the front, back or top. Consult your owner's manual to figure out how to get in, or do an Internet search for your type of dryer.
Put a chalk mark on the drum, next to the rubber belt encircling it. This is so that when you begin turning the drum, you will know when it's made a full turn.
The belt will be attached to a wheel on a motor. Different models have different configurations, but all have a point at which the belt comes away from the drum to be pulled by the motor. Find that position, so that you can see the underside of the belt, the part that is otherwise pressed against the drum.
Hold your can of belt dressing over the underside of the belt. With your free hand, begin slowly turning the drum while spraying the belt dressing onto the underside of the belt. Spray in an even, consistent stream as the belt moves under the stream. Continue turning the drum until you see your chalk mark come back to the top position.
Put the dryer back together. The squeak should now be gone.