What role does the storm play in the development of the plot?

In the opening scene of the play, the royal court of Milan is violently tossed during a strong, unexpected tempest, which threatens to sink the ship and kill all of the passengers aboard. Shakespeare's audience believed that the weather and nature, in general, was a reflection of human relationships and emotions. Therefore, the tempest at the beginning of the play corresponds to Prospero and his daughter's unjust exile, when Antonio usurped his position to become the Duke of Milan. The violent tempest also serves as the catalyst of the play, which results in the royal court being shipwrecked on Prospero's magical island, where he masterminds a plan to retake his title from Antonio while allowing Ferdinand to fall in love with his daughter Miranda. The audience discovers that the tempest was not a natural occurrence and that Prospero gave Ariel direction to purposely create the tempest. Fortunately, Ariel's tempest serves its purpose and the royal court is shipwrecked on Prospero's magical island, where he gets revenge for being exiled and is eventually restored to his rightful position as Duke of Milan.

The violent sea storm—the tempest of the title—is the catalyst for what happens in the play. It's the storm that brings the royal party to Prospero's island, where all the action takes place. And it was Prospero who whipped-up the storm in the first place using his magical powers. Moreover, the storm is important as it sets the scene for what's about to follow. The storm's violent tempestuousness is a reflection of the seething anger and resentment that Prospero still feels over this overthrow and subsequent exile. By creating the storm, Prospero is showing the royal party on board ship that his supernatural powers are much stronger than the merely earthly power they enjoy back in Italy. This is his way of showing them that he's the one who's now in charge; he's the one who controls their destinies.