"Smart casual" is the way Princess prefers to label its general dress code, and smart it was on my cruise. Even the pool-wear seemed to exude "smartness" and class. There are two formal nights on a seven-night cruise: lots of beaded gowns for the ladies and tuxes for the men, although cocktail dresses and dark suits are perfectly acceptable. No swimwear, jeans, tank tops or shorts are allowed in the restaurants at dinnertime.
A pair of smart trousers and a shirt (no tie necesary) is the norm for most meal nights. A suite or black tie is appropriate for the 'formal' nights. But you can always stay casual if you prefer and eat in the 24 hour restaurant.