You are assigned either late (8:30) or early (6:00) seating. You may make a request, but it cannot be guaranteed. You then are assigned a rotation through the 3 restaurants on the ship - Triton's/Lumierre, Animator's Palate, and Parrot Cay.
When you board your KTTW card will show you your rotation (designated by the first letter of the restaurant) and you will have "tickets" which have your seating time and assigned table number on them.
You just need to show up at the right place at the right time, OR eat at Topsider's, room service, "snack" places, etc.
The only time you have to make reservations (and unless you have the REAS package you cannot do it until you board) is if you would like to try Palo's, which has an extra cover charge of $10 per person.