During the day, all I ever wear is my bathing suit with shorts and a tank top or t-shirt over it. It's also good to bring a sweatshirt or other long-sleeved shirt. All cruise lines that I've been on expect people to dress up a little bit. When I cruise, I dine in the formal dining room almost every night, since it's the only time I really spend with my parents. Girls on the other hand, have a lot more options.
Girls can wear anything, from nice khaki or black pants to long skirts to short skirts to dresses.
I find that people have the most questions about formal nights, and in a way these are the hardest questions to answer because there is so much variation. I can tell you what is most commonly seen though. Guys go for pants and a sport coat, and many go for tuxes. You can bring your own tux, or most ships have a Formalities shop that rents them out. A lot of guys don't like going to formal night because they feel kinda dorky wearing a tux.