You just turn up but you may have to wait if you pick very busy time.
I've heard that if you do pick a busy time, they give you a pager so that you can go and have a drink in a bar while you wait rather than stand queuing up outside the restaurant. Howeve I don't think the range of the pager is that far, so you may not be able to roam that far.
You can make a reservation,if you like the table it maybe possible to keep it for the week.
On the Saphire princess last June you had to book every evening and after trying the pager route we always booked after that. The range is good though as we went to a bar way over on the other side of the ship and the things went off before the estimated time so all was well but the concept of anything dinning has changed as on the Star princess in 2005 you didnt need to book.