What is the external conflict in Animal Farm?

We might very well phrase the external conflict in Animal Farm in Marxist terms. Marx believed that the proletariat, or working class, was essential for the wealth enjoyed by the capitalist bourgeois class. It's clear in the speech by Old Major at the beginning of the novel that the animals are representative of the proletariat and Jones is the capitalist. The animals provide the wealth for Farmer Jones through their work and their very lives. Old Major says,

Man is the only creature that consumes without producing. He does not give milk, he does not lay eggs, he is too weak to pull the plow, he cannot run fast enough to catch rabbits. Yet he is lord to all the animals. He sets them to work, he gives the bare minimum that will prevent them from starving, and the rest he keeps for himself.

This is precisely the same argument Marx made about the social classes in Europe in the 1800s. He thought the capitalists and the bourgeoisie lived off and exploited the labor of the proletariat. Marx predicted there would be a breaking point and the proletariat would eventually rise up and stage a revolution against the upper classes. Under his hypothesis, industrialized workers would rebel, overthrowing the bourgeoisie and forming a new classless society. Unfortunately, neither Marx nor Old Major could conceive of a new ruling class (in history, the Bolsheviks; in Animal Farm, the pigs) who would continue to exploit the working class while telling them that their new society was far better than the old capitalist regime.

The main external conflict throughout Orwell's novella is Man vs. Society, which is illustrated by the conflict between animals and humans. Mr. Jones is an extension of society and neglects the animals by continually forgetting to feed them. Early in the novella, old Major speaks about the dangers of interacting with humans and vehemently declares that all humans are enemies. Following old Major's death, the pigs develop his ideas into a system of thought known as Animalism and create laws preventing animals from interacting and behaving like humans. After initially expelling Mr. Jones during the Rebellion, he returns with his men and the animals repel his attack during the Battle of the Cowshed. When Napoleon usurps power, he begins trading with humans and is taken advantage of by Mr. Frederick, who buys timber with forged bank notes from Animal Farm. The next morning, Mr. Frederick and his men attack the farm and end up destroying the windmill.
Another external conflict depicted throughout the novella is Man vs. Nature. Under Napoleon's tyrannical reign, the animals suffer from the harsh conditions during the winter. They are malnourished and the unfavorable weather prevents them from having successful harvests. The building materials needed for the windmill are also located at the bottom of a deep quarry. Animals like Boxer struggle to lift and carry the stones to the appropriate location to construct the windmill.
Man vs. Man conflict is illustrated by Napoleon and Snowball's struggle for authority and power on the farm. In an effort to destroy his political rival, Napoleon raises nine enormous dogs and trains them to attack on command. During an assembly, Napoleon instructs the dogs to attack Snowball and they end up chasing him off the farm. For the remainder of the novella, Napoleon uses Snowball as a scapegoat when anything on the farm goes wrong.

In Animal Farm, there is an external conflict between animals and humans. Orwell introduces this conflict early in the story: in Chapter One, for example, Old Major expresses this conflict when he assembles the animals in the barn to warn them about the "tyranny of human beings." He also teaches them a song from his childhood called Beasts of England which inspires them to unite against the humans. This conflict is resolved in Chapter Two when the animals successfully overthrow Mr Jones and take control of the farm for themselves.
Later in the story, this external conflict is again expressed through the animals' relationships with their human neighbours, Frederick and Pilkington. This is best shown when both men bid for a pile of seasoned timber which Napoleon wants to sell. In the end, Napoleon sells the timber to Frederick but is double-crossed when he realises that the bank notes are nothing more than forgeries (in Chapter Eight). This conflict is resolved through battle and the animals are, once again, victorious.