A main conflict in "Shooting an Elephant" is between the narrator's hatred of the colonial and imperialist system he is part of and his concomitant hatred of the Burmese people. As he so memorably puts it, he "bitterly" despises being part of the British police state in Burma, but

I thought that the greatest joy in the world would be to drive a bayonet into a Buddhist priest’s guts.

He recognizes that imperialism has created the hatred the Burmese feel towards their British overlords, but he also resents that hatred, which manifests itself in attempts to jeer at, harm, passively resist, and discredit the English.

The conflict that arises from this situation is between common sense and the British need to keep up an appearance of invulnerability. This is best expressed when the narrator feels compelled to save face by shooting and killing a harmless elephant. He knows it is cruel—the elephant will die slowly—and wasteful—the elephant poses no threat at this point—but he does it because the Burmese crowd expects him to. Orwell shows through this incident that the imperial system is irrational, cruel, and debilitating.

In "Shooting an Elephant," Orwell highlights a number of conflicts. Firstly, there is a conflict between Orwell and the native Burmese people. This is made clear in the first paragraph when he says that he was "hated by large numbers of people." These people, the Burmese, would jeer, shout insults, and in one case, tripped Orwell up while he was playing football.

Secondly, there is a conflict between Orwell and the British Empire. He believes, for example, that imperialism is an "evil thing," having seen first-hand the treatment of Burmese prisoners, which he describes in the second paragraph.

Finally, Orwell experiences an internal conflict which is shown through his dilemma over whether or not to shoot the elephant. On the one hand, Orwell does not want to kill the animal because he knows that it will eventually calm down and that he does not have the correct equipment. But, on the other hand, if he does not shoot the elephant, he will be humiliated and the Burmese will never take him seriously again.

The main conflicts in "Shooting An Elephant" revolve around a colonial policeman in British governed Burma.  He has a great deal of difficulty with the people he must protect.  He believes that they harbor a resentment and a distinct prejudice towards him.

This resentment is built around the fact that the British treat their colonized subjects as inferior, a fact that disturbs the policeman greatly.

One of the conflicts arises from prejudice and tolerance

"The colonial policeman has a duty towards the job, towards the empire, and this in turn requires treating the locals as inferiors."

Which leads to understanding the conflict of culture clash between the British rulers and the native people.

"The first is the ethical difference setting the narrator, as a representative of the West, apart from the native Burmese, who belong to the local village-culture and live in a pre-industrial world from which the West itself has long since emerged."

The narrator has a conflict of conscience.

"The narrator's moral conscience appears in the moment when the corpse of the Burmese crushed by the elephant comes to his attention; the narrator says that the man lay sprawled in a crucified posture,"

The last conflict comes from the action of order and disorder.  The elephant escaping is a sign of disorder, the policeman is a representative of order. 

"which is why Orwell’s narrator cannot avoid the unpleasant duty of shooting the elephant."