When you're indignant, you're angry about an unfair situation. If you discovered that a teacher gave ten extra points on a test to all students who sat in the front row, you'd be indignant.
Indignant is from Latin indignus "unworthy," and it refers to anger based on unworthy or unfair behavior rather than merely injury to one's own interests. You may be angry, even furious, if someone shoves you, but you are indignant if the shove is directed at someone weak or helpless. The related noun is indignation, and something that arouses indignation is an indignity.