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. Again, girls have a lot of variety in what they can wear. My favorite thing to do is bring past homecoming and prom dresses with me. The cruise is a perfect opportunity to wear things you never thought you'd be able to again.