Hi, Denise!
There is no real required formal dinner aboard a Disney cruise; the only exception is if you choose to dine at one of the adults-only restaurants (
Palo on the Wonder and Magic, the Dream and the Fantasy, and
Remy on the Dream and the Fantasy).
While formal attire is certainly appropriate and accepted, it is by no means enforced; the dressiest I think I have ever gotten on a Disney cruise is trousers and a nice blouse. Some people do take the "formal dress night" suggestion very seriously; some of your fellow diners will be in walking shorts or capris.
The Disney Cruise Line is wonderful about catering to the vacation needs of its Guests, and making sure that each of its voyages is fun and memorable for everyone who sails The Cast and Crew Members who work on the ships are certainly more interested in your enjoyment than your attire.
Bon voyage!
See you later, celebrator,
-Deborah B.