As Duke of Milan, Propero's first duty should have been to govern but because of his interest in magic, he gave over his authority to his brother. Antonio went further in his desire for power by usurping Propero and setting him and the baby Miranda adrift. Both men misuse power.
On the island, he used his power to free the spirit, Ariel, and he took care of Caliban until he tried to rape Miranda. Both became servants to Prospero.
Despite the fact that he was a powerful magician, he did not have he power to get off the island. With the assistance of Ariel, he caused the tempest which brought the ship with the king's party to island. There are several reasons for this action.
The ship is his way off the island but more importantly he wants to get Miranda off the island so that she may live a "normal" life. He loves his daughter very much and knows it is time for her to leave.
He also wants to revenge the wrongs committed by his brother and his allies but chooses in the end to forgive them. Here Prospero shows that he has learned that mercy is better than vengence, a powerful lesson.
Prospero breaks his staff of power and when he returns to Milan, it is suggested that he has learned how to use power properly and will be a good administer.
Consider the roles that magic and social hierarchy play in the context of The Tempest. When Ferdinand and Miranda have their wedding, the spirits gather around them to bless the union. They command nature to watch over the new couple. Magic here is used as a form of natural power in the benefit of the two lovers. The magic is used properly to seal and protect the state of love between Ferdinand and Miranda.
On the other hand, power and authority are often used improperly by Prospero who keeps Caliban and Ariel in his service. Prospero killed Sycorax and took Caliban as his slave, even though Caliban claims that he has natural rights to the island. Similarly, Prospero keeps Ariel as his servant, and although Prospero has promised several times to free Ariel, he always finds another use for the sprite (i.e. creating the storm that drives Alonso's boat to shore).