Possible problems:
1. Bad battery
2. Bad charging circuit
You can perform a power cycle if the battery is removable (shut the laptop down, remove the battery, remove the charger, hold the power button down for 5 seconds). You can re-insert the battery and charger, then power it back up to check out whether it's charging or not.
Chances are that you will either need to replace the battery (first resort) or replace the motherboard.