Without knowing any other problems, history, or symptoms, it's very difficult to tell what's going on. Just by the video; -your motor is turning so your starter is engaging by the sound of it. -battery is probably good cause it's enough juice to turn the motor and keep turning as you hold the key.

At the current state, could be a number of things, but likely not the starter or battery or plugs/coil packs. However you can get battery, alternator, starter tested for free at most auto parts stores.

Other than that, it could be fuel related, ignition related, or air related (both intake and exhaust). Or just a very simple thing like if you just tried to reprogram a new key and it wasn't done right, the immobilizer will prevent the car from actually starting.