Assuming you're requesting dinner reservations:
First, decide your first, second and third choices of night and time to eat. When you turn in your cruise documents at the terminal, they will hand you a mini-Navigator which will show where the reservations will be taken. Have one person from your party go to where the reservations are being taken immediately upon boarding. Upon arrival in the res room, there will be a chart showing which days/times still have space available. If your first choice is taken, you will still have two alternatives. There will be three tables taking reservations, each handling a couple of days, with clearly-posted signs. When you get to the table, ask for the day/time you wish and give them your KTTW card and stateroom number. They will hand you a small card with your reservation on it. If all of your days/times are already full when you get there, you can ask to be put on a waiting list for cancellations.
If making reservations for afternoon tea or brunch, there will be a separate line for these and the chances of getting in are even better than dinner.
All that being said, I have made reservations on two cruises now with no problem at all. Good luck.