From the factory this is wired from current path 15 (switched 12V from your ignition switch) to the cluster. From there it goes to through a resistor to the + side of the BATTERY light (LED). From there it goes to the infamous "Blue Wire" which is the exciter wire on the alternator.
The idea is that, when the key is on, but engine off, the alternator isn't making voltage and the LED finds ground through the field winding which turns the LED on and warns you your battery isn't charging. When the engine is on, the alternator is making voltage and that voltage opposes the +12 from the ignition switch and the LED is off.
The voltage through the LED also serves another purpose. It's to excite the alternator and have it start making voltage.
These alternators are typically self-exciting about 2000-2500 RPM, so even without that voltage the alternator should start working if you blip the throttle to 3000 RPM or so.
Try leaving it disconnected and starting your car. Note the voltage on your console gauge (or use a voltmeter at the battery). Rev the car above 3000 RPM for a second or two. Does the voltage suddenly jump up? That indicates your alternator is self-exciting and you don't need to connect this wire at all.
If you have the wire connected directly to the field winding from the battery then power is going through the field winding at all times when the alternator isn't operating. That's bad for the battery and likely bad for the field winding as well, since there is nothing to limit the current. If you need to do this, use switched power by jumping from the + side of the coil to the alternator field wiring terminal. Place a 120 ohm, 1/2 watt resistor in line with the wire.

Thomas

My peanut bulb in the harness-plug suggestion (your other thread) did not work?