The red badge of courage refers to the wounds that the soldiers suffered in war. These wounds were viewed as courage Eva courageous acts because it meant that they stayed and fought against the enemy. Obviously, the red stands for the bloodshed.

In traditional literary works, courage was defined as a soldier who rushed head long into battle, showing no fear.  War was idealized and the men who fought in it were hereos - consider Odysseus as a prominent example.  However, soldiers understand that war is not so clear-cut as this, and neither is courage.  Sometimes courage does mean having the courage to run away, to hide, to protect yourself when all else seems to be failing.  Courage is also about living up to your duties and responsibilities.  Because each individual has his own duties, no one can call another "courageous".  It must be a personal "badge".

His self-pride was now entirely restored. In the shade of its flourishing growth he stood with braced and self-confident legs, and since nothing could now be discovered he did not shrink from an encounter with the eyes of judges, and allowed no thoughts of his own to keep him from an attitude of manfulness.

This badge of courage is "red" because red is the color of passion and of blood.  The red represents the struggle of war and the struggle to do what is brave.