Hi Denise!
For the vast majority of Disney restaurants, you should feel completely comfortable wearing what you wore to the parks for dinner (or any meal, for that matter). At some of the Signature Dining Establishments (these are the fine dining restaurants, and include: California Grille, Artist Point, Jiko, Flying Fish Cafe, Narcoossee's, etc.) there is a "business casual" dress code. However, even these guidelines are quite liberal. For men, khakis, slacks, jeans and dress shorts are acceptable along with collared shirts. For ladies, capris, skirts, dress shorts and jeans are all fine. Swimwear, hats for gentleman, cut offs and tank tops are not permitted. You can see the dress code by clicking on a particular restaurant on the Disney website and scrolling down to "dress code". Click
HERE for the California Grille's dress guidelines as an example. Again, business casual applies only to a small number of signature dining restaurants.
As for whether you'd like to bring dress up clothes, it's not necessary, but rather entirely up to you. I definitely recommend leaving the "little black dress" at home! You might want to bring some cute capris and sandals though! You'll find that some folks like to get a bit "dolled up" for dinner, especially later in the evening, while others prefer to enjoy their meal in their comfy park clothes. Either is absolutely fine.
Bon Appetit!