How exactly does diplomatic expulsion work?

Diplomatic expulsion occurs when the government of a host nation, such as the United States, declares a foreign diplomat "Persona Non Grata."

Persona Non Grata simply means "an unwelcome person." Governments normally do this when foreign diplomats engage in "activities outside their official capacity", a phrase that is used as diplomatic shorthand for espionage. They also sometimes expel diplomats in response to their own diplomats being expelled, or when the foreign diplomat engages in a serious crime.

Once declared Persona Non Grata, the foreign diplomat usually has about ten days to leave the country.