Removing the battery

Hello,

When should I remove the battery to prolong its life, before putting the AC power or after? I keep on removing the battery of my notebook but recently, "I noticed that when I removed it, it couldn't be turned on just by an AC power. I don't know if it about the charger or what. An sometimes, I remove the battery after the laptop is turned on. Is it Alright. Hope you understand what I'm trying to say and I really need your help. Thank you.