On short Caribbean cruises they call it "Dress Your Best" or "Wear Your Best" or "Casual Chic" instead of Formal Night. That means a step up from your typical tropical dinner wear. They mean collared shirts, dresses, skirts, blouses and pantsuits. You will see jackets, tuxedos/formal wear, sports coats, blazers and long evening gowns on longer cruises and more so on longer European and Asian cruises.
However, as some have posted, you'll see everything including jeans (without holes and tears) and flip flops and unless more than 75% of your body is not covered where it should be, you will not be turned away (i.e. thong bikinis, tank tops, hot pants, etc). Happy Cruising!