Allegory

As Orwell wrote Animal Farm as an allegory—a symbolic representation of real events—many characters and events in the novel symbolize individuals or groups in the Russian Revolution and the Soviet Union. Some of the characters who symbolize individuals or groups in Soviet society include Mr. Jones (the Russian Tsar and the old aristocratic order); Old Major (Karl Marx and Vladimir Lenin); Napoleon (Stalin); Snowball (Trotsky); Squealer (the press); the dogs (the secret police); and Moses the Raven (organized religion). Nearly all of the other animals represent the working class and Soviet peasants.

Animal Farm as Satire

A satire is a work which uses humour, irony or wit to highlight the vices, follies and pretensions of individuals, institutions, communities or ideas. Animal Farm satirises the breakdown of political ideology and the misuse of power, and does so in the ingenious form of a beast fable. The major players are animals but their failings are all too   recognisably human. They begin with an idealistic attempt to form a new society, liberated from the tyranny of humans and founded on the principle of equality and freedom for everyone, but it all goes wrong as the pigs take over. Backed up by the brute power of the dogs, they appropriate all manner of comforts and even luxuries for themselves, while reducing the the other animals to the same condition of slavery that they suffered under humans.
Orwell's point that the pigs are really just the same as the human tyrants they replaced is underlined in the famous ending to the novel, as the pigs mingle with humans to the extent that it becomes impossible to distinguish between them:
The creatures outside looked from pig to man, and from man to pig, and from pig to man again; but already it was impossible to say which was which.(chapter 10)
The novel, then, exposes the perversion of political ideals and the corruption of power which occur all too regularly in human societies. Most obviously perhaps, it functions as an attack on Stalinist Russia, where the original Communist Revolution degenerated into war, interior power struggles and the emergence of a grim totalitarian regime under Josef Stalin. However, the satire of Animal Farm is not tied to any one time or place. Its lessons are universal, and conveyed in memorable fashion, and as such it endures as a powerful and relevant literary work. 
George Orwell used the real-life Russian Revolution, which resulted in the U.S.S.R. and many human-rights violations by the corrupt leader Joseph Stalin, as the template for Animal Farm. The satire comes in the very obvious power-grabs by Napoleon, who represents Stalin, and how the animals are prevented from questioning him by violence and propaganda: [Napoleon] announced that from now on the Sunday-morning Meetings would come to an end... In future all questions relating to the working of the farm would be settled by a special committee of pigs, presided over by himself. These would meet in private and afterwards communicate their decisions to the others.
Most of the plot echoes real-life events from the Battle of the Cowshed (the Russian Civil War and battles against Germany) to Napoleon's use of trained dogs to kill animals he deems subversive (the KGB and secret police who turned in Russian citizens who voiced "subversive" opinions). By limiting the setting to a farm, and showing the deliberate lies told as propaganda, Orwell both condemns and satirizes the Marxist ideals that drove the Russian Revolution.
The most direct use of satire in Animal Farm is through the use of characters as representations of the Russian Revolution. Old Major represents Karl Marx, the father of Communism, and Napoleon and Snowball represent Joseph Stalin and Leon Trotsky, respectively. By showcasing their actions as they affect the farm -- Snowball genuinely wants to improve their lives, while Napoleon only wants personal power -- Orwell shows how personal gain works against the ideals of Marxism. Another good example is Squealer, who represents the government propaganda machine of Soviet Russia, more importantly the newspaper Pravda; by controlling the flow of public information, censoring important events, and appealing to fear, Squealer convinces the other animals that the pigs need to be in charge for the common good. Similarly, in Soviet Russia, all news and information flowed through government-approved channels, and dissenters were hunted down and killed for daring to disagree with the accepted line of thought.