He is suddenly declared a traitor and escaped from Animal Farm with Napoleon's snarling dogs (remember Bluebell's puppies?) hard on his heels.
Snowball represents Léon Trotsky, a militant communist leader who opposed Stalin's regime. He was the founder of the Fourth International League (which contested Stalin's Third International League though never managing to overpower it.)
In the power struggle which ensued, Trotsky (or 'Trotski' or 'Trotzsky') was "chased" from the government in 1924, then excluded from the Communist Party in 1927. He was banished from the USSR four years later after being sent into exile in central Asia. He was ultimately assassiated in Stalin's political purge.
Snowball is branded a traitor by Napoleon and chased off the farm into the hedges by Bluebell's puppies, who are now Napoleon's own personal guards. Over time, Snowball grows into this mysterious anti-revolutionary boogeyman for Napoleon's regime, as Napoleon blames Snowball and his followers for everything that goes wrong on the farm. Napoleon even rewrites history and says that it was he who came up with the idea of the windmill, and that Snowball only copied him. The animals, who, with the exception of Benjamin, seem to have really short memories, agree with Napoleon and assist with blotting out the positive aspects of Snowball's leadership.
In the book, Snowball is supposed to represent Leon Trotsky. Trotsky believed more in trade unions than in absolute power invested in one person, and he and Lenin soon became rivals. In the period after Lenin's death in 1924, Trotsky was outmaneuvered and forced into exile. Just like the animals who favored Snowball, Trotsky's followers were either converted to Stalinism or purged. While we are never sure of what happened to Snowball in the story, Trotsky was assassinated in Mexico.