Banishment is a diplomatic action which can be delivered to one's prisoners who are also direct subjects.
Banishing a character costs 10 piety. With 
Conclave active, banishment may trigger a council vote.
Banishing a courtier
Banishing a courtier seizes their wealth and forces them to flee. However, the character retains their artifacts.
Banishing a direct vassal
Banishing a direct vassal forces them to abdicate all titles to their heir(s). They flee and retain their wealth.
Banishing vassals invokes tyranny. The penalty depends on the titles of the character you're banishing, generally -10 tyranny per title. Banishing brothers as a Muslim does not incur tyranny.
Banishment reasons
Some opinion modifiers that allow free imprisonment also allow banishment without a piety or tyranny cost. These include most opinion modifiers related to murder, murder attempts, or mutilation.