Spielberg uses a black and white sepia film stock in Schindler's List to give verisimilitude to the film, to take the edge off the bloodshed, and to strike a contrast between awareness of the Holocaust and apathy.
The film begins in color. A prayer is spoken and the yellow flame of a candle dissolves to the smoke of a train. The shift in color connotes a shift in time, from present to past. The news is in black and white. So is World War II footage. In fact, most of the greatest films of all-time (except perhaps 2001: A Space Odyssey) are filmed in black and white. Visually and psychologically, viewers trust black and white. Spielberg's cinematography strikes the perfect balance (known as chiaroscuro).
The black and white also serves to stylize the violence. The point-blank executions are grizzly, to be sure, but filming them in black and white makes them less visceral. The blood is black, instead of red. As such, the bloodshed is less gratuitous.
Schindler is awakened to the Holocaust during the liquidation of the ghetto when he sees the little girl in the red coat. She is an Innocent, and her red coat is symbolic of the aforementioned blood that has been spilled. Now, the blood is on Schindler's hands, as he is aligned with the Nazis. He soon realizes that he must save his workers from the same fate as the girl. Later, Oskar will see the girl in the red coat's charred remains in the concentration camps. She is a reminder of the victims he could have saved (if he had awakened to the horrors sooner).
The film ends in color as it shifts back to the present. The Schindler Jews place stones on his grave as an homage to his sacrifice.