The "Captain" that Whitman is a reference to Abraham Lincoln, whom Whitman was very fond of. The opening line of "Oh Captain, My Captain" is "O CAPTAIN! my Captain! our fearful trip is done;" in this he is referring to the Civil War. He is alluded to throughout the entire poem. "Where on the deck my Captain lies fallen cold and dead" is reference to the fact that the President had been assassinated shortly before the end of the Civil War.
