Any time a car won't start - and the starter motor clearly isn't spinning the engine - turn the headlights ON with the ignition key in the "ON" position or the pushbutton START stuff in the normal "car on" mode. If the headlights are close to normal brightness, the battery and connections are probably okay. If the lights are dim, or are visibly loosing brightness rapidly, the battery and/or connections suck.
Typical "battery good, wires good, starter won't spin though" problems in cars:
1: transmission Neutral/Park sense switch bad. Wiggling the shift lever may "fix" this. Very common bug. Just run the lever to "Neutral" and try starting. If this "fixes" the car, it means a) the switch is loose/falling off and has lost it's calibration, or b) the switch and/or linkage are busting.
2: brake pedal switch failing, especially on cars with Genesis style pushbutton start.
3: the solenoid has two electromagnet coils typically. If one fails, the remaining one isn't enough to "slam" the mechanical parts inside. Ergo the switch never gets closed and the starter motor never gets power to spin. On stick-shift cars, putting the transmission into 1st or reverse and gently nudging the vehicle in the same direction slightly turns the whole engine... which may be enough to help the "slam" part work. The "slam" has to drive the starter motor's small gear into the teeth of the flywheel; if the teeth happen to line up it's an easy slam. If they don't line up, the "slam" has to nudge-turn the engine a little.
4: no response from the starter/solenoid at all - but the headlights stay bright - can mean the "immobilizer" part of the alarm has tripped, preventing engine startup.
A "clicking" starter motor means the solenoid (the thing that acts like a switch for the really high amperage starter motor) is switching ON-OFF-ON-OFF. When it goes ON, that connects the battery to the starter motor. At this point, a weak battery or bad connections would suddenly manifest themselves - facing the big amperage draw of the motor - and the system voltage drops as a result. When the voltage drops, the solenoid may no longer have enough voltage to work so it disengages (it's basically a big electromagnet that drives a big switch) which in turn disconnects the motor... removing that big amperage load. Now the battery "recovers" and the voltage rises, re-energizing the solenoid. Click-click-click.
Bad battery cables, or bad cables from the chassis to the engine block (ground cable), lead to these symptoms as does as a weak battery.
Sometimes a physically hot starter motor/solenoid assembly acts this way too; thermal expansion of the plunger part makes it jam in the electromagnet coil body. Tapping the starter/solenoid body sometimes makes it kick in enough to start the vehicle. Very common especially on engines with exhaust pipes crammed against the side of the motor (to fit in a narrow engine bay) so the hot exhaust pipes bake the starter too. Some vehicles actually had small fans blowing air on the starter! When that fan wears out, the starter bakes and the engine won't start.
mike c.