Thank you so much for playing! The enemy you're talking about must be the giant head that goes looping down the corridor, right? There are two of them in the game, and you can kill them by shooting them a lot, but it's not mandatory.
In the corridors of the first giant head, you need to find a duct to get a key for a door that will lead to the last room of the game. In that room, you'll also need another key to open the door that ends the demo.