It is really your choice on how you would like to use the second monitor, typically they would not have the same thing on, but they can do this (in my roommate days we had four monitors set up showing the same movie so that we all had a good view in the living room).
Really, the second monitor becomes an extension of the first one, to an extent. One is your primary monitor (think start bar and desktop) and the others are just extra space. You can drag a document or program from one to the other and your mouse pointer just kind of jumps through the air onto the second screen. For instance, I often work with documents that need changing - one screen will have the primary document body, next to that I keep the bibliography and appendix that will later attach to that document, and on the third screen I keep the internet (well not all the internet, just my access to it) for quick checks. It surprisingly saves a lot of time when you don't have to flick between tabs etc.
My only problem is that I start to feel a little claustrophobic when surrounded by too many screens for too long
Sort of, except you can't really take an image or movie and let it stretch across the two screens. Think of it more as having two different laptops next to each other - now imagine the ability to have the mouse pointer from one laptop go across the screen borders and onto the adjacent screen. You can then also drag a program or anything you are working on from the one laptop right onto the other. Now just imagine the second laptop to simply be a monitor.
