When you turn your key in the ignition, or push the start button, battery power flows through the starter relay or solenoid and starter motor. The starter motor engages a gear (usually called a pinion) with the flywheel ring gear, spinning the flywheel and getting the crankshaft moving up and down in the cylinders. At the right moment, the spark plugs are energized and fire, compressing the air/fuel mixture into a spark, thereby turning over the engine.
If the sound is a true grinding noise, it indicates the starter motor is not engaging the flywheel correctly. It can also indicate that there is damage to the flywheel ring gear teeth. Replacing the starter motor is usually the correct solution.
Another possibility is that your fuel pump has a problem. A dying fuel pump will cause an air bubble to form in the system, reducing the pressure in the pump and sometimes making it difficult or even impossible for the pump to properly inject the fuel.
Other possibilities include a loose wheel bearing or worn shock or stabilizer bushing. Unibody cars that get driven on salted winter roads often develop rusted suspension pickup points, which also make a clunking noise.