That sounds like a starter solenoid. What you would here is the gate inside the solenoid slamming shut when power is applied. That would make a single 'click' noise. It shoudl close the starting circuit between the battery and the starter. However, if the contact in the solenoid is bad, the circuit won't close properly and the starter gets no power.
That is a common problem. If it's the solenoid, you should be able to turn the key or switch on and off a bunch of times and the solenoid should eventually complete the circuit.
If it's the starter, then a few raps of the handle of a screwdriver should get that moving as well. I would pursue solenoid first.