Here's what you should investigate:
* Battery Voltage: Use a multimeter to test the battery voltage both when the car is off and when the engine is running. A voltage below 12.2V when off indicates a weak battery. A voltage that doesn't rise to around 13.5-14.5V when the engine is running indicates a problem with the alternator.
* Battery Condition: Have your battery tested at an auto parts store. They can assess its charge, cranking amps, and overall health. A load test will determine if the battery can deliver the high current the starter needs.
* Starter Motor: While less likely given the headlight dimming, a failing starter motor is possible. However, a bad starter usually results in a *click* sound rather than no engagement at all. Given the battery seems to be the main culprit, address that first.
* Connections: Check all battery terminals and cables for corrosion and tightness. Loose or corroded connections can increase resistance and prevent the starter from getting sufficient power. Clean any corrosion and tighten all connections.
In short: Start with the battery. Have it tested, check the voltage, and clean the terminals. If the battery is weak or faulty, replace it. If the alternator isn't charging properly, that also needs to be addressed. Only if these checks are negative should you suspect the starter motor.