Banishing a courtier
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.
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.