non formal nights, slacks (dockers), button or polo sport shirt, think country club casual. This is simply being polite to your dining companions.
formal nights, depends on how you personally address formal. You will encounter everything from Tuxedo's to suites or sport jacket, dress shirt and tie. You will not be out of place wearing any of the three.
You will encounter everything across the entire clothing spectrum, what you will actually see in the dining room depends on how the ship enforces their dining room dress code.