elephant also symbolizes a rebellion against the rule of the British.  The Burmese people clamor for the elephant's death just as they long to see the death of the huge British Empire's rule in their country.

3.  The idea of the elephant dying slowly in a "world remote," demonstrates the death of British ideals and ultimate power.  Ironically, Orwell sees the British way of life being destroyed not in Britain where humans might mourn their loss but in a far away land where Britain tried to impose its customs upon others.

I would also answer that the elephant is important because the narrator's actions and situation with the elephant mirror colonial/imperial England's relationship with the Burmanese. The elephant is a work animal in India and Burma, and for the colonial power, the Burmanese are also 'work animals'.

As the above comment mentions, the elephant is potentially dangerous, as are any people oppressed by another, however, by the time the narrator reacts, the danger is no longer present.

The elephant is the key to the narrative persona's difficulties with his position in the Burmese society. He shoots the elephant for all of the wrong reasons, and, because the elephant is an innocent victim of his ego, the elephant comes to represent the effects of the tension between colonizer and colonized in Burma.

The elephant is a potential danger, but the narrator acts long after the danger has passed. The narrator knows there is no benefit for him in shooting the elephant beyond acting as he is expected. The elephant is also valuable. He sacrifices something beautiful, almost human, and valuable simply to appease the needs of his ego. The elephant's physical size is also important, for without the weapon, the narrator would be no match for it.

I'd add one thing to the answers above, both true: the elephant also symbolizes his shame. As a colonist, he occupies a place that he has no right to occupy. He is neither superior nor especially fit to govern. The elephant is like the Burmese people. Large, natural, and apparently needing to be controlled. The entire horrendous situation could have been avoided, and yet the narrative persona mindlessly follows the expectations of others, even though he is hopelessly incompetent. He knows there is no need to act--the elephant is already beginning to calm down when he shoots it. But he doesn't feel as if he can get out of the situation. He has power, but, as Jamie says, not conscience. So he kills an innocent in order to avoid looking like a fool. In killing the elephant, however, he reveals himself a fool. He is the worst kind of authority--someone who acts because they feel compelled to act, not because of an inner conviction, or a commitment to right action.

The previous answer is correct, but I would like to add that the elephant is also symbolic of the narrator's 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, invoking all of the poignant and rich symbolism that the term ‘‘crucified’’ offers. The elephant, too, especially in its pain-wracked death, evokes in the narrator feelings of terrible pity, not soothed by his knowledge that he acted within the law. Law, indeed, opposes conscience in ‘‘Shooting an Elephant.’’ The brute fact of Empire, thoroughly institutionalized, is irreconcilable with the individual’s moral analysis of the situation."

There is more on the themes, symbolism, and other literary elements of Orwell's story at the link below.

In "Shooting an Elephant", the Elephant represents the working man since in India and Burma, the elephant is a work animal. It can also be seen to represent the role of the Burmanese to the colonial power - in this analogy; the Burmanese would be the colonial power over the elephant. At the end of the story, the animal takes on definite human characteristics as it dies.

The elephant represents the narrator's conscience, which he tries to ignore (it is the proverbial "elephant in the room.")

Here is an excerpt on the theme and symbolism of the elephant from eNotes. You can learn more about this and other elements of Orwell's gripping tale by visiting the link below.

"His official position, rather than his moral disposition, compels the narrator to act in the way that he does, so as to uphold his office precisely by keeping the native Burmese in their subordinate and dependent place. As a colonial official, the narrator must not let himself become a spectacle before the native crowds. Not shooting the elephant would make him seem like a coward, so he shoots the elephant. 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, invoking all of the poignant and rich symbolism that the term ‘‘crucified’’ offers. The elephant, too, especially in its pain-wracked death, evokes in the narrator feelings of terrible pity, not soothed by his knowledge that he acted within the law. Law, indeed, opposes conscience in ‘‘Shooting an Elephant.’’ The brute fact of Empire, thoroughly institutionalized, is irreconcilable with the individual’s moral analysis of the situation."

We are never completely certain of the extent to which the story "Shooting an Elephant" is autobiographical. Orwell was actually an imperial policeman in Burma, and he uses the first person and an autobiographical tone throughout. So the narrator, in some sense, at least, is Orwell himself. Beyond that, the narrator represents the corrupting effects of British imperialism on the colonizers themselves, as well as the colonized. The narrator is placed in a situation where he is forced, as he sees it, by his relationship with the Burmese people to shoot the elephant, which he does not want to do. But they expect him to, and he does it to avoid, in his words, "looking a fool." He feels insecure and hated (he discusses this at length in the beginning of the story) and the only way he can avoid looking foolish is by living up to their expectations of him. Imperialism is, at the end of the day, a violent enterprise, and the narrator, insecure in his power in the face of a mob, must satisfy them by behaving violently and against his wishes.