What is the importance of Snowball being expelled in Animal Farm?
The ousting of Snowball paves the way for Napoleon to become absolute dictator of Animal Farm, just as Josef Stalin because unrivaled leader of the Soviet Union after the ousting of Trotsky. While Snowball was on the farm, he was an effective counterweight to Napoleon. He was not immune to grabbing privileges for the pigs, but at the same time Snowball worked tirelessly for the common good. It was he who proposed building a windmill, an idea Napoleon would pooh-pooh.
When the dogs run Snowball off the farm at Napoleon's command, Napoleon is emboldened to take on more and more power. Without Snowball as a check, Napoleon rules supreme and unchallenged. Under Napoleon, the idea of making the lives of the regular animals better increasingly fades. Napoleon is happy to make sure his will is always paramount.
The expulsion of Snowball from the farm by Napoleon is important for many reasons. Firstly, it shows the way that Napoleon identifies that Snowball, with his eloquence and rhetoric, is a threat to his own power. He sees that Snowball is winning the hearts and minds of the animals on the farm and recognises that he has to do something about this if he is going to carry on being in power.
Secondly, note the parallels between Napoleon and Snowball and the real life situation of Stalin and Trotsky. Napoleon chooses to work secretively, behind the scenes, in a covert way that involves stealth and surprise, whereas Snowball bases his power on his popular support and his ability to inspire and capture people with his ideas. In the end, it is Napoleon's power that shows itself to be stronger.
Lastly, the manner in which Snowball is chased out of the Farm shows the way that the democratic principles of the Farm are slowly but surely eroded by the use of authoritarian power and violence:
At this there was a terrible baying sound outside, and nine enormous dogs wearing brass-studded collars came bounding into the barn. They dashed straight for Snowball, who only sprang from his place just in time to escape their snapping jaws.
The Farm began its existence as a place where everybody had equal rights. Now we see that it is a place where those who have power decide what rights the animals get. This is the start of the decline of equality.