Yes, it does hurt. You will have your jaw wired shut until it heals up, which is a pain in the butt and you will have bone pain, which is some of the most intense pain there is.
However, it will heal, and they will give you pain meds, and you can eat lots of stuff through a straw.
If your lower jaw is much too jutting or much too short, you'll have other pain for the rest of your life from the muscles in the jaw being poorly aligned. Plus, it will make your smile much more perfect.
Typically, they don't break a jaw for anyone unless the underbite or overbite is severe. Before agreeing to have it done, I would talk to ANOTHER orthodontist and get their opinion. It's no small procedure, so if you don't actually NEED it done, standard braces can fix almost all of your problems.
My childhood orthodontist wanted to break my jaw and my parents didn't go through with it -- a few years later my teeth and jaw fixed itself and my teeth are nearly perfect WITHOUT messing with them. Sometimes you get bad advice, so double check.