Sounds like the starter solinoid to me too. Means a new starter. The next time it won't turn over, hit the starter with a hammer or a heavy wrench, and see if that gets it to turn over. If it does, you need a new starter.
Why does this work? I've done it before on an old mustang I used to have, but I've no idea why it works. Anyone?
(look at how I didn't hijack the thread I got this quote from)
I think if it's stuck and it can't turn the shock of hitting it will loosen it up.
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To answer your question more specifically Tiny: The starter motor has a moving part in it that shuttles back and forth when power is applied to start the engine. The Bendix (as it's referred) engages when you turn the keyswitch to the "start" position and releases when you allow the switch to return.
With age and abuse (mud, water, and especially salt), the bendix can become stuck in the starter housing or stuck on the shaft and will not deploy when you try to engage the starter. Of course, the starter is off more than it's on, so it ususally get stuck in the off position. If you use a hammer and beat on the side of the starter, you can ususally break it lose. You generally know it's the starter because when you hear the "clicking" sound when you are trying to start the engine, nothing happens and the motor will not crank. The clicking sound is the solenoid engaging but if the starter is stuck, then it will not crank over.
The starter solenoid also has a moving part located inside the unit and it can jam as well. Sometimes the internal contacts on the solenoid become corroded and get stuck. A lighter tap is suggested for the solenoid or you can simply jump it out using a screw driver if you're real slick.
Good story on the solenoid - I taught my wife that trick about five years ago. She was picking up her father one night in her '68 Mustang. I was previously having a problem with the solenoid so I showed her how to bypass it if it gave her any trouble. Of course that night it would not start.. So, she opens the hood and grabs the screw driver form the glove box and started the car... (My father-in-law still hasn't stopped talking about that event.)