What causes the battle between King Arthur and Sir Mordred to begin?

On a literal level, when Arthur leaves the court to fight Lancelot over his presumed affair with Guenivere, he places Mordred on the British throne. Mordred begins to form allegiances with false promises, and the people resist Arthur's return. At this level, the war is a power grab, and Arthur fights the civil war to regain his throne.
On a more symbolic level, we sense that Mordred stands for an inner force, caused by human failing (Arthur's affair with Morgause). Camelot was superior to all other nations and could not be defeated in battle. Its weakness is internal, like a classical "hamartia" (fatal flaw). Mordred represents the consequence of Arthur's and Camelot's weakness.
Without Launcelot or Gawain, Arthur returns to claim his throne profoundly lacking in his most trusted companions. He was warned to not fight until Launcelot could rejoin Arthur, but in the fatal truce ceremony, a snake accidentally crosses a knight's foot. Instinctively, he draws his sword to kill the snake, but this is misread as a sign of aggression and a betrayal of the peace terms. War breaks out at this point.
This seemingly meaningless accident leads to the brutal battle that leaves only a handful of Camelot's knights standing. In this way, we can read the war as a series of accidents and betrayals dating back to Arthur's affair, Launcelot and Guenivere's affair, Gawain's wounds from fighting Launcelot, Launcelot's delay in coming to Arthur's aid, Mordred's political contrivances rather than honor, and a snake.

Mordred is Arthur's illegitimate son. And Merlin prophesies that, if the boy is allowed to grow up, he will one day destroy both Arthur and his kingdom. That being the case, the wily old wizard urges Arthur to round up all the babies born on May Day—Mordred's birthday—and have them put out to sea. Yet the infant Mordred somehow manages to escape the ensuing massacre and lives to fight another day. When he grows to manhood he becomes his father's most implacable foe, hell-bent on destroying Arthur and taking his kingdom for himself.
As well as being a traitor, Mordred routinely flouts the code of chivalry to which knights are expected to adhere. For instance, he kills Sir Lamorak by stabbing him in the back, an act considered deeply dishonorable for any knight of the realm. Mordred is so power-crazed, so obsessed with grabbing power, that he unilaterally declares himself king and tries to marry the queen. And all this before he actually kills Arthur at the Battle of Salisbury Plain.