Running a car without engine oil will result in damage to the engine. The check engine and check oil lights should come on when there is no oil in the vehicle, and the vehicle should be stopped immediately.
Without oil, the engine overheats, and damage, such as pistons and rings fusing to the cylinder walls and the engine block cracking, will occur. The cylinders will lose compression, coolant will leak and the engine will likely need to be replaced. Even if the engine block and heads are not damaged, it will likely cost less to replace the engine than it will to rebuild it.