Animal Farm by George Orwell serves as a warning to anyone who tries to rationalize the use of power and suggest that "some (animals) are more equal than others." He purposefully set the book in an environment that could serve various possible notions of the abuse of power, although it mirrors the actions of individuals and events during the Russian Revolution of 1917. His characters can be compared to Karl Marx (Old Major), Leon Trotsky (Snowball), and Joseph Stalin (Napoleon), for example, and some of the events parallel the real power struggle between Stalin and Trotsky, even to the point of Trotsky (and Snowball) being used as a scapegoat after having been expelled.
However, using animals means that the story can be, as the subtitle suggests, a fable or "fairy story" without removing any of the significance. This also allows Orwell to reach a far wider readership. There are those who enjoy the story without any historical significance (some school children do not know the background to the book). Then there are many people in the western world who appreciate its universal quality and can relate the events to other world leaders, even in the twenty-first century. There are also those who relate it to their own personal struggle within a country with an autocratic government.
The symbols from the story include the farm itself, the windmill, the seven commandments, and even Old Major's skull, which is dug up after his death as a symbol and a reminder of what the animals' so-called freedom stands for. (In 1924, Lenin's body was put on display in Moscow.)
The farm can be seen as symbolic of Russia and Communist rule or can represent any society that is susceptible to the abuse of power. One abuser is ultimately replaced by another as the pigs become indistinguishable from the humans.
The building of the windmill is comparable to the modernization projects in Russia after the revolution and reveals the real exploitation that is possible, and how easily manipulated people (or animals) are. The pigs (Napoleon) only serve themselves, but are able to persuade the animals that their actions (and ridiculously hard work) will benefit the greater good. Blaming Snowball for sabotaging the windmill and their plans ensures that the animals become more dependent on Napoleon in their fight against Snowball, their supposed common enemy.
The seven commandments (which compare to Karl Marx's manifesto) are inspiring and promise a great future, but gradually their potential as the ultimate symbol of freedom (the socialist concept that Communism promised) is eroded as they are changed to suit the pigs. Self-serving governments take advantage of citizens who are often less educated and blinded by empty promises to the point that citizens overlook the signs that the very government that won their trust is corrupted by the influence of power.
This information will hopefully allow you to decide on a symbol and explain its importance and relevance.