Whenever your dryer refuses to run, first be sure it is plugged in and that the receptacle works. If the receptacle doesn't work, the circuit may have overloaded--check and reset its circuit breaker. Be aware that most electric dryers don't plug into standard 120-volt receptacles--they operate on 240 volts.
Clothes dryer problems usually involve the heat source or the motor that turns the clothes drum. Clothes don't dry because it doesn't get hot enough or the drum doesn't rotate.
If a gas dryer doesn't dry properly, the first thing to check is the pilot light or igniter--it may not be lighting the burner (read your owner's manual for proper re-lighting techniques). The second thing to check is the dryer's vent--be sure it isn't clogged with lint.
If the dryer doesn't start, check the door switch. Press and release it. Door switches break from time to time. You can remove the switch, test it, and replace it if needed and you have the time and skills. Otherwise, call a repairperson.
If the dryer runs, but the drum doesn't tumble, call a repair service. The machine probably has a broken motor, belt, or drum turning mechanism. Also call a repairperson if the dryer rotates but doesn't heat at all.
Check the vent regularly for lint buildup; a blockage can keep clothes from drying.