My car will not start. Battery has power but this TRW 14103304 part (looks like a big fuse) is making a clicking noise when i try and crank my car. Then when i turn the key all the way over the dash lights go out or go very dim.

Check the Crank/Cam and coil fuse, one of them blew.

the alternator isn't charging battery that would account for loss of power and also clicking when try to start , try a jump start if it starts then you know its the battery so with engine running check the alternator charge rate with a voltmeter across the battery terminals anything below 13 volts then check the alternator belt is tight enough , good condition etc , then recheck charge rate if still low fit new alternator

• sure it won't crank the engine , starter motor just clicks !
• try using jumper cables, and crank, off a running car.?
• Fuses all good.? test them?
• battery terminals clean, never green?
• A/T transmission? did you wiggle shifter, as you crank it.?
• a/T trans, did you try starting in neutral?
• M/T trans did you press clutch pedal harder as you cranked it over? (repeat say 3 times, see if this makes it crank)
• try a good battery from a good car? in your car.
• charge your battery. using a battery charger..
• Does hot wiring the starter solenoid , make it crank over?

Can you jumper the starter solenoid terminals to make it crank ?
Is the security light staying on or blinking ? Did you try cranking it in neutral instead of park ?

No, not very normal, if it won't start. The clicking may be from your starter relay-there is an electrical problem that needs diagnosing. Does it crank over and not start, or won't even crank? Just a guess, but you may have a fuse link burnt out. Check if your fuses have power to them. When the key is in on,
every fuse on the car should have power to them. Use a test light and check.

Perhaps the thick power cable to the starter is broken. Is it a little click like a relay or more of a clonk like a solenoid? The starter often has the solenoid built in. It's this clonk that connects the heavy power to the motor. You don't usually notice this clonk because it's immediately followed by engine cranking.

So two possible causes;
1. Big power wire to the starter
2. Solenoid not firing.

Clicking noise is the sound of the starter being bad and not having enough power to turn the motor over. Replace the starter.

try the starter solenoid or the starter relay....be sure all the starter/ignition fuses are good

batterys can look great on a guage and have no juice try getting a boost and also check the master and assessory fuses under the hood in the fuse pannel
sounds like you blew a master fuse