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This sounds like an issue with your charger. I think replacing the charger will resolve your problem.
If you unplug the charger from your laptop, look inside the round circle that goes inside of the notebook. Inside this circle is a wire in the center, if that wire is bent or broken than the charger will behave as you have described, it provides power but does not charge the battery.
If you can see the wire in there is bent, unplug from wall outlet (of course) and you could try and straighten it, usually it will break though and you will end up needing to replace the charger.
I am about 95% sure a new charger would resolve your problem...
There is something you can try though to see if this could be a software problem on your laptop and not a broken/loose wire in your ac adapter.
Shutdown the computer completely so that it is off, then plug in the notebook for several hours and see if you are able to get your battery to fully charge. If the computer will charge while it is turned off, but not while it is on...then we may need to look at some things in your operating system.