Why is my computer not charging?

Bad battery or logic board

click on the apple, then click about tis mac,>more info. In the left colum click power and the battery staus will be there

Your battery may be faulty. To check that, I would get the free iStat for Mac widget and check to see if everything is what it should be. Perhaps your main logic board is faulty, maybe not. If you are paranoid, run the Apple Hardware Test (Or diagnostics). Since you're running Snow Leopard, put in the Applications disc that came with your computer and hold down the "D" key during boot. You should see the diagnostics screen and begin testing.

However, the not charging green light could also indicate that your computer is using so much power, that the battery will not charge for safety reasons.