Your voltage regulator (mounted on the back of the alternator) is what turns off the BATT light. When you turn your key to ON, 12V flows from the battery to the BATT light and then a wire (I think it's green with a white tracer) goes from the light to the alternator, providing the ground for the light. When you start the engine, the voltage regulator turns OFF the ground for the BATT light and it goes out.
Check to see of the alternator is charging by verifying voltage at the "B" terminal on the back of the alternator (large red wire) and then check voltage at the battery. You should see around 13.6-14 volts in both places. If you get good voltage at the alt but not at the battery, you Mega fuse may be blown. It's located on the rear of the power distribution box. As previously stated, you alternator must be grounded to work, but this usually isn't an issue as it's bolted to the engine.
If you don't own a VOM (Volt OHM Meter), you can buy a cheap one at any auto parts store. A very useful tool for diagnosing electrical problems.