1992 Nissan Sentra INSTRUMENT PANEL LIGHT COMES ON D/D OFF

I BOUGHT A BRAND NEW BATTERY CAUSE MY OTHER BATTERY WAS DEAD, AND WHEN I INSTALLED IT AND DROVE DOWN THE ROAD, I NOTICED THAT THE D/D OFF LIGHT WENT ON, AND THE NEXT MORNING MY BATTERY WAS DEAD BUT NOTHING WAS LEFT ON, ANYWAYS I DO NOT KNOW WHAT THAT D/D OFF LIGHT MEANS SO THAT I MAY CORRECT THE PROBLEM AND REPAIR IT, COULD YOU PLEASE HELP ME WITH THIS MATTER, I DO NOT KNOW WHAT THAT LIGHT MEANS, I WOULD REALLY APPRECIATE IT, THANK YOU