On a typical 7 night Caribbean cruise you will have 2 formal nights.
the first formal night is the Captain's Gala Welcome Aboard party - this is the dressier of the two formal nights. It is usually the night of your first FULL day at sea. The second formal night is the Captain's Farewell - usually the night of your LAST FULL day at sea.
As for dress, on Princess the majority of the women wear long gowns or dressess, long dressy skirts w/sparkly or silky tops, dressy cocktail dress, dressy dinner suits, or dressy platazzo pants suits. If your ship has the Midnight Champagne Waterfall and Party on the last formal night, GO! it is great fun, the MaitreD and his staff build this huge champagne waterfall, the mood is very festive, champagne, streamers and you get to party with the staff.