There should be a specific reason for each animal.
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The pigs represent leaders, specifically Soviet/Stalinist leaders. I think the negative connotations of pigs also come into play. Pigs want everything, they are greedy, and they are also known to be highly intelligent creatures. The pigs use their intelligence to manipulate the masses (other animals). Orwell decides to give the other animals different traits as found in communist, and to an extent democratic, society.
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There was already literary precedent for a farmyard pig to be a leader. In 1937 Walter R. Brooks began writing the Freddy the Pig childrens stories. A total of 26 were written well into the 60's. Freddy was quite a capable pig, a porcine raconteur. He was an author, an editor, the Ambassador to Mars, the list goes on.
Many of the social themes that Orwell uses in Animal Farm were explored less graphically by Brooks in Freddy the Pig stories. The comparisons though are there for you to read. In fact no academic study of Animal Farm is complete without regarding the Brooksian influence of Freddy the Pig upon Orwell.
One of the reasons Animal Farm was so widely accepted both in the United States and the UK was the huge popularity of Walter R Brooks childrens series and the tales of Freddy the Pig.
It was not then incongruous for Orwell to have animals as protagonists, heroes or villains or for adults to embrace Animal Farm as literature; they had already been reading Freddy the Pig by Walter R Brooks for 30 years. Walter R Brooks also created Francis the Talking Mule and Mr. Ed. Both animals were far smarter than the humans who fed them.