In Animal Farm Snowball is branded a traitor and chased off the farm. This is part of an effort by Napoleon to use Snowball as a scapegoat and to get rid of a rival. Snowball's exile happens in Chapter 5.
Snowball and Napoleon continue to debate about the windmill. The windmill was Snowball's idea and he wants it because it will help the farm. Napoleon is opposed to it because it will slow down food production. Napoleon also wants to stockpile weapons while Snowball wants to spread the revolution to other farms. The windmill, however, becomes the central issue of debate. Finally, the idea is put to a vote. It looks like the windmill is going to pass. Before the vote can happen, Napoleon calls out the nine puppies (now grown dogs) he has been raising. They chase Snowball off the farm. Napoleon then imposes new rules and takes over the farm--choosing to build the windmill anyway.
Snowball is used throughout the rest of the book as a scapegoat for anything that goes wrong on the farm. The whereabouts of Snowball are actually unknown, and it is doubtful he ever puts a hoof back on the farm. However, the demonizing of Snowball provides the Pigs, especially Napoleon through Squealer, a way to put fear into the animals on the farm.
In the animated movie of Animal Farm (1950), the sequence of Snowball's exile gives the impression that the dogs assassinated Snowball. The second movie (1999) which was shot with live action and CGI- he escapes the dogs, but Napoleon claims he is dead. Both are deviations from the novel.
Snowball's character leaving the farm is also symbolic. Snowball is the thinker of the two warring pigs. He believes in innovation and is an idealist. Rather than using force, Snowball relies on debate to convince those to follow him. He is ultimately undermined by brute force. Likewise, rhetorical debate regarding communism is undermined by military control, which leads to despotism.