Orwell's "Shooting an Elephant" is ultimately his metaphor for giving in to peer pressure. Orwell only shoots the elephant because it is what is expected of him. He gives in to pressure from those around him and does what, under other circumstances, he ordinarily would not choose to do.
He shoots the elephant three times out of mercy. He references the fact that after he shoots it the first time, the elephant was obviously hurt but had not moved. After the second shot, the elephant falls down but is still not dead. The third shot is necessary to end the suffering of the animal. Orwell immediately regrets his decision after the first shot but follows through to the third out of a feeling of mercy.