Your dryer's drum tumbles clothes in order to increase the flow of air circulating around and through them. It's difficult for clothes to dry if the heat isn't properly circulated and impossible for them to get dry if the drum isn't turning.
Before trying to figure out why your dryer isn't working, turn off the power to your dryer by unplugging it or by taking the fuse out of the fuse box. If you have a newer home it probably has breaker switches instead of fuses. Locate the dryer's breaker on the breaker panel and turn it off.
Generally, there are four components in your dryer that could be causing the problem and each is easily diagnosed.
Assuming your dryer's motor is working, the first component to check is the belt between the drum and the motor. To access the belt, remove the front panel of the dryer's cabinet. The belt should loop over the drum, under the idler pulley wheel and around the motor's drive pulley. If the belt is in the correct position and everything looks good, check it for small cracks. Take a small section of belt, approximately 2-3 inches, and bend it between your thumb and index finger. If the belt is cracked, it should be replaced. As a rule, the belt on your dryer should be replaced every 5-8 years.
The second component to check is the idler pulley wheel. The pulley is attached to a moveable arm and applies tension to the belt. Slip the belt off the pulley and give the pulley a spin - it should spin smoothly and not wobble. If the pulley doesn't spin smoothly or wobbles, it needs to be replaced. You can replace just the pulley on some dryer models, but others require an arm assembly as well.
The pulley and belt usually wear at the same rate so if you replace one component you need to replace the other as well. Never operate your dryer with a damaged pulley - it puts unnecessary strain on the motor pulley and could damage the motor.
The third component you should check is the motor. Use a multimeter to check the motor's terminals and ground connection. If the motor fails either of these tests it should be replaced.
The fourth and final component to check is the door switch, a one-inch plastic part with metal terminals. The switch prevents the dryer from operating when the dryer door is open. When you close the door, it pushes a small button on the dryer's front panel. The button activates the door switch and the dryer starts. When you open the door, the button releases, turns off the switch and the dryer stops.
Removing the switch on some dryers is easy - you just pry it out. Other dryers need to have the front panel removed. Test the door switch for continuity using an ohmmeter. Improper readings indicate the switch is defective and it should be replaced.
You may need a service technician if you find this repair difficult to do yourself or if you don't have an ohmmeter or a multimeter. A service technician can perform the test for you and advise which parts you need to replace.