Generator light on dash

Turned the car off last evening and noticed the red generator light on the dash was on very faintly. I started the car back up and it went off immediately but came on again when I shut it down. It's a very faint light that you would miss in bright sunlight. I don't drive the car much at night so maybe it's been going on for a while now, I dunno. Unlike most of the people on here I'm fairly green when it comes to working on cars so I'm not exactly sure where to start. Any help would be appreciated.