The first thing to do when your dryer won't start is to check the breaker at the electrical panel. Make sure the breaker has not tripped. If so, then reset breaker and try starting your machine.
If you still have no starting function take a look at the door switch, is it intact? If it is not functioning electrically or mechanically you will need to replace it and it will require getting inside the dryer cabinet. You will need a multi-meter to test for on/off continuity. If you do not know how to get into your dryer it would be advisable to call an expert technician.
At this point, another thing to check is the thermal fuse. This is located inside your dryer by the blower wheel housing and is a non resetting fuse. To test you will need a multi-meter. These tools can be purchased at any hardware or box store. If test fail replace and your dryer should start up. Depending on age of the dryer you could also have something happening with the timer, control board or the motor. For these diagnostics you should consult an experienced technician.
Other tips on dryers are when they will not start, drum will not tumble, taking too long to dry or how to get a dryer repaired.