Although this is a rare occurrence nowadays, it still happens.
If the engine cranks, but car doesn't start:
Bad fuel pump: A faulty fuel pump won't deliver fuel to the engine, even if you've just filled up.
Bad spark plugs: Without hot sparks to ignite the fuel charge in each cylinder, there is no combustion and no expansion of the burning mixture. Thus, the engine will not start.
Bad timing belt: If the timing belt stretches or breaks, the engine will not run. Moreover, continuing to crank the engine may bring the valves into contact with the piston, causing serious engine damage.
Bad ignition coil: Without a functional coil to step up the voltage, there can be no spark. No spark means no start.
Faulty cam or crank sensor: If the sensors are faulty they will either provide no information to the ignition system, or will provide incorrect data. Either way, the engine will not fire.
If nothing happens after you turn the key:
Bad battery or terminals: Batteries have a finite lifespan, and terminals can loosen or corrode.
Bad starter: This should be obvious from the response to the key. If nothing happens at all when you turn the key, the starter may have reached the end of its life.
Ignition problem: This could be due to faulty contacts in the ignition switch, or in the wiring.
If there is a grinding noise when you turn the key:
Starter is bad: Do not persist in cranking the engine.
What do you do now?
Bring your vehicle to your local mechanic. The mechanic will check the ignition system, fuel system, and charging system, and then look for any mechanical damage. The mechanic may also try an alternative fuel source, like a quick shot of brake or carburetor cleaner into the throttle body, or a quick shot of liquid fuel via a vacuum line. If the car starts momentarily and then dies, that indicates that you have a fuel delivery issue.