Exile usually refers to leaving one's native country. James Joyce lived in exile from Ireland. It can also mean someone who has been forced out of their native country - by political opponents for example. You can also go into exile or become an exile of your own free will. It is not the same as emigrate. There is also internal exile; it usually refers to a form of political punishment. This was used to describe Russian dissidents under Communism, who were exiled to Siberia. Expel means that you have been thrown out of an organisation or body to which you belong. Commonly used about schoolchildren who have been expelled from school for misbehaviour. Or you can be expelled from a political organisation. But you would not be expelled from work or your job - the equivalent of that is to be sacked or fired. Oust is to overthrow someone from a position of power. The rebel forces ousted the king. The king was ousted by his courtiers.

be exiled expel and bar (someone) from their native country, typically for political or punitive reasons oust -drive out or expel (someone) from a position or place -deprive (someone) of or exclude (someone) from possession of something