I agree with Lin, what you have is a Split-leaf Philodendron. It's not really the same as Swiss Cheese Philo which has smaller leaves with holes. Here's mine:
I also agree that you can pot it up, turn that long stem around into the pot again, and have it carry on. But you probably already know, these plants get huge. The name "Monstera" is a clue here. It will keep on creeping along like that and trying to climb out of the pot. Turning the pot regularly to keep it growing towards the light will help it make a curve and stay in a pot for a while, but eventually you're going to want a big, sturdy stake in the pot with burlap fastened to it, so the thing can climb.
