For Narcoossee's the men will need to wear a collared shirt with dress shorts or pants. Women can wear a dress, a blouse or nice sleeveless shirt (like would be appropriate in an office... not a tank top or spaghetti straps) with dress pants/shorts or a skirt. No t-shirts, jeans, cargoes, or flip flops for anyone, and only closed shoes for men.
I've only ever worn a dress, and only gone with women but I've never seen anyone who was dressed more casually, so either nobody tried or they were turned away. If you don't have time to change before then I'd suggest a nice summery dress (I wear maxi dresses at WDW all the time) for the ladies and dress shorts and a lightweight short sleeved button-down for the gents. You can be dressed well and still be comfortable in the parks.