When you are given your Key To The Workd card on check-in it will have your dining rotation on it. Each night you will be assigned to one of the Main Dining Rooms. Your servers will come to each MDR with you, so you will have the same servers the whole time.
Different length cruises have different rotations. You can do a search for rotations as lots of people list them.
You can request a specific rotation in advance, however Disney will let you know it is only a request and not guaranteed. Lately a few people have reported that rotation requests are not preferred as rotations may change, so depending on what you want, your request may be better phrased as something like "we would like to be in Animator's Palette on Pirate Night" or similar.