If a car will not turn over but the lights and radio still work, the problem is either a bad starter motor or a faulty ignition switch. A clicking noise when turning the key usually indicates a problem with the starter motor. No clicking sound and no sound from the engine can indicate a bad ignition switch. In both cases, the solution is to replace the faulty items.
There are two main reasons for a car not starting. One is lack of spark, the other is lack of fuel. Lack of spark can be caused by a dead battery, a dead starter, a faulty ignition or bad spark plugs. A lack of fuel is usually the fuel pump or problems with the EFI system. A dead battery or alternator is fairly easy to diagnose because the rest of the electronic systems will either be dead or dim. A clicking sound means the starter motor is trying to engage, but failing. No clicking sound means that it is either the ignition or the starter solenoid.
Another possibility, unique to automatic transmissions, is that the car is in a gear other than park. Most automatics will not start in any other gear except park. In that case, the lights and radio would work, but the car would not turn over.