It looks like your white MacBook is a 2008 model (1181). Knowing which version of OS X would help, but do you still have your Leopard disk set that came with the MacBook?
And does your MacBook work OK when plugged into just AC (not running on battery)? The battery may need replacing. Sometimes that model will run on just AC without the battery and sometimes it won't. You might want to remove the battery and see if you can start it just on AC power.