@ missraffix3, Is your charger made by Apple, or is it a clone? The green LED means that full power is being delivered, and the non-Apple chargers do not communicate with the MacBook's charging system, so they always deliver full power. The Apple charger adjusts to what the MacBook needs, and only delivers full power when its needed. The Apple charger is better, but an 'always on' green LED on a non-Apple charger is nothing to worry about. If it is an Apple charger, I suggest taking it and your laptop to the Genius Bar to see what's wrong.
@ Everyone, If saving battery cycles means anything to you, it is better to run your MacBooks from the charger whenever you can, and when you do need to run off of the battery, go ahead and use the battery until it runs down, so that you'll always do a FULL recharge. The LiPo batteries aren't harmed by partial recharges, but every recharge cycle counts, even if its only 10%.
The MacBook will shut-down before the battery is drawn down below its safety limit.