I have a Mustang that the battery light stays on.
I changed the battery and the alternator was checked by Autozone, they said it was good.
I need help
The battery light works by connecting the battery's plus side (through the ignition switch!) to the alternator's positive output. It's probably the oddest piece of circuitry in a car, but that's why it glows when either isn't working. And why it has to be an incandescant globe (but you could use two LEDs to indicate which has failed).
It sounds like a wiring problem, but not necessarily with the light itself. I'd check the engine, alternator and battery are properly grounded (though that should create ignition problems) and that the charging cable is whole. If the alternator is failing to charge the battery, you will probably notice the engine note drop sharply when you turn on something high-current, such as the headlights.
First off, you may want a 2nd opinon on the alternator. If both battery and alternator are good, then its either a wiring or ECU problem. Check the contacts on the battery terminals/connectors to make sure they're not loose, dirty, or corroded. Then follow the ground wire (the black one) back to the chassis of the car to make sure it has a solid ground..