I have an 04 Accord EX. When you drive down the road with the headlights on they are dim, then if you let off the accelerator then the lights brighten back up. Every time you push the accelerator again the lights dim. Does anyone have any ideas about what might be causing this problem?

It's likely the voltage regulator, but have the complete alternator tested. If confirmed by tests, some will replace just the voltage regulator, others the complete alternator.