I have got apple macbook pro unibody , my charger light is blinking and not charging battery while my laptop is on but when i shut down my laptop its connecting and disconnecting for 10 seconds interval

SOURCE: macbook Battery Problems

Hi Molle
To answer your last question first. It is best if possible to use the mains power when you are using your macBook and only use the battery when you cannot connect to mains.

MacBook batteries have a life span based on the number of charge cycles the battery has been through. The battery has a max of 300 cycles before it starts to loss its charge capability. SO you need to determine the health of the battery is it chargeing etc. You can do this via the apple menu and 'about this mac'

http://support.apple.com/kb/HT1519

Check the health of the battery as the battery could be faulty if it has only cycled a few times or it may have gone over the 300 and is now lossing its charge in which case you'll need to get a new one.

Hope that helps

Paul

SOURCE: macbook pro start up problem

First remove the battery, then plug it into your AC adapter. Hold the power button for 5 seconds or until you either hear and beep or the activity light on the front of your laptop lights up. Start up normally. 

SOURCE: Macbook Pro just shuts down

This sometimes happens when you have it plugged in to an electrical socket as well. I have recently been having this problem and I have a brand new battery, so it's likely the logic board. If you're still having the problem, and the monitor image shows weird fragmentation on startup, it's likely to be the logic board that needs to be replaced.