I would assume that either charging circuitry has failed or battery is dead. Easiest way to diagnose which one is to find same battery (friend with same battery model) and put it in your computer.
If you cannot change that battery, than charging circuit is culprit here. For that you will need to take your Mac for repairs. If another battery works, just order replacement.
SMC Reset
If you contact Apple, the first thing they'll tell you to do is an SMC reset. It's easy, just turn off the machine. Then hold in the power button until you hear this horrible long beep sound, then let go of the power button. Should have to hold that button in for about 15 seconds. Now, that may help you, but for the various times I've been told to try that, I've never actually noticed doing that help me. But, Apple will tell you to do that, so you may as well try it now.
Power Management Firmware
Another thing is there have been various firmware updates around power management for my MacBook, which is far older than yours. You should be certain no such updates exist. You'll have to boot in to OSX and use the Software Update utility to check.
Power Brick
I won't be too surprised if you message back saying you've got this far without a change. I've had the kind of behavior you've had before, actually a couple times. One time, it was my power adapter. I've used my (many years old now) MacBook a TON and I managed to wear out my power brick/adapter. You can easily enough test that, by taking your MacBook in to an Apple store (or some place that has them on display). Just plug in their power to your machine and see if that seems to cut it. Of course that's not a totally certain test, but if it's looking good, then you should ask Apple for a replacement power brick. Be mindful about the power capacity of the brick you try too. For example a MacBook Air brick isn't meant to source enough power for a 17" MacBook Pro.
Battery
I think it's unlikely to be your battery itself, but it could be. If you've been using that thing profound amounts, your battery may be pretty battered already (heh, almost a pun). Go in to the System Info app (linked to from About This Mac under the menu bar at the top). Click on your battery and check the charge cycle count. If it's getting near 300, then your battery is at its end of life and you should ask Apple for a replacement.
Motherboard
Failing that stuff, I'd really suspect your motherboard to be the issue. This is where you arrange to go to an authorized Apple Service center (or the Apple Store nearest you). They'll fix it up, given the age of that machine, you should have minimal hassle.
Like I said, I've dealt with this kind of stuff a few times with my MacBook. I've been a little annoyed about having power problems (or any for that matter) with my machine, but I've found Apple to be extremely good at service. The worst Apple service I had was through a 3rd party servicer. I've also had some good 3rd party ones too.
Hope that helps ya.
Best Answer
Find yourself an Apple Store, show them the issue, they may replace the battery right there if it's in warranty and the battery is easily removed (unlike the newest line of MacBook Pros). I've had them swap out my battery on an out of warranty machine too.
I believe the basic test for you is:
Of course, Apple has a fancy battery testing tool that they will reboot your computer into once you're there in person, so I implore you to find an Apple Store, an Apple Certified tech who won't charge you for warranty repair, or contact Apple Directly about the issue.
For more info on what counts as a "cycle" read this http://www.apple.com/batteries/
Similar issue discussed on Apple's Discussion forums: http://discussions.apple.com/thread.jspa?threadID=1296373