It's a time-limited demo (15 minutes), but you can play it multiple times. You can go anywhere in the play area but there are barriers restricting how far you can actually travel, so you can't visit the whole city. I'd say there are maybe five or six city blocks to move around in.
I had to play it twice, the second time to try and do the stuff I didn't have time to do the first time.
For anyone who has played the full game, the demo starts just before the part where Batman meets Catwoman in the court house and Joker tries to shoot Batman.