Let's take it by animal species from the top to the bottom.
The pigs (the leaders of the pack) are confident, selfish, strong-willed, deceitful, and in need of constant pleasure and comfort. The correspond very well to the corrupt leaders/dictators you hear about around the world.
The dogs (the police/enforcers) are like the sidekicks. They are loyal, cruel strong arms of the government who believe what the leaders tell them and will enforce the state rules no matter who gets hurt. (Think the Nazi SS)
The horses (Boxer and Clover) are like the loyal workers. They believe what they are told because they are too dumb to think for themselves. They will work harder than anyone because they are good hearted, but they don't see how they are being used.
The Raven (which represents the church) is holier than thou and hypocritical. He sidles up to whomever is in power and spouts cliches about heaven. He supports which ever side brings him the most comfort.
The Donkey is the skeptic, but he is also selfish. He is smart and knows what is going on, but he wants to stay out of it and mind his own business.
The chickens and the sheep are the general public, pushed and pulled by every wind of doctrine.
Old Major- is the idealistic believer in equality and ultimately the purest form of communism. He is loyal and honest, and wants the best for his "people". He corresponds to Karl Marx in history.
Napoleon - is ruthless and brutal and has a lust for power. Much like his historical counterpart Stalin, he cares only for himself and will sell out anyone to give himself more power and status.
Snowball- is a type for Trotsky in old Russia. He believes that the pigs should be in charge, but he wants to bring forth communism through education, not force.
Squealer- is the "spin doctor" for the pigs. He manipulates the people to believe in the world state.
The Dogs- are the military arm of the government. They are mindless "right-hand men" of the pigs. They have no thoughts of their own, but simply carry out the brutal commands of their leaders. Think the Nazi SS.
Boxer and Clover - are the true believers. They work very hard to make everyone equal, but they are too stupid to understand that they are being taken advantage of. They are much like the everyday people under communism.
The donkey- is smart but stubborn. He knows what is going on, but he doesn't want to get involved. I would equate him to those during WWII that turned a blind eye to the holocaust.
The Raven- is hypocritical and self-indlulgent much like Orwell's view of the church. He sidles up to whomever is in power and makes promises of a better life, so the leaders will keep him happy.
The sheep and the chickens- are regular joes.