Prospero was the Duke of Milan. His wife had died around the time Prospero's daughter Miranda turned 3. Miranda is the princess of Milan
While Prospero studied, Miranda was raised by caretakers. Miranda had many caretakes because of her nobility.
Prospero gives his brother Antonio the permission take government power. Prospero would rather study instead.
"The government I cast upon my brother, And to my state grew stranger, being transported And rapt in secret studies" (I.ii.5).
Antonio uses this opportunity to alter the government against Prospero.
While Prospero studied alone, Antonio began to replace Prospero's loyal men and manipulated them to serve him. Prospero had put his trust into Antonio but the power had made his brother into a power hungry monster.
Antonio pays the King of Naples who also hated Prospero so they could get rid of Prospero. Antonio at midnight opens the gates to Milan and lets the Naple army in.
Prospero was forced from his position as king. Antonio knew he could not kill him due to his popularity in the city so he sent him and his daughter off by boat hoping that he would die at sea. A nobleman named Gonzalo provided the two of them with food, clothing, and books.