Possibly a bad battery connector. Here's how to change it: MacBook Core 2 Duo Battery Connector Replacement
According to if your's is an energy saver, here's the part: MacBook Battery Connector/Sleep Sensor (Energy Star)
Be sure to check the exact model number before buying.
Provides battery contacts and sleep sensor for logic boards with Energy Star printed at the second RAM slot. May or may not have a liquid damage indicator patch. — Currently out of stock.