Symbolism, Imagery, Allegory
Like the storm in King Lear, the tempest that opens our play is full of symbolic meaning.
When Prospero uses magic to whip up a storm that shipwrecks the King of Naples on the island, the tempest seems like a very physical manifestation of Prospero's anger and his suffering, which has been eating at him for the past twelve years. Big surprise there, right?
Although the tempest (like Prospero's anger) is definitely powerful enough to cause a shipwreck, no real harm is actually done. Prospero wants to teach Alonso and Antonio a lesson, but the fact is that he doesn't kill anybody or cause permanent damage to the ship or its inhabitants. Unlike the notorious storm-whipper-uppers in Macbeth (that would be the Weird Sisters), Prospero is not an evil guy. He's bitter, controlling, and wants some payback for losing his dukedom, but, ultimately, Prospero forgives the men who once betrayed him.
We also want to point out how the tempest is associated with social upheaval. You probably noticed that as the crew and passengers are being tossed around on deck, panic sets in and quite a lot of trash talking goes down after Duke Antonio tries barking orders at the crew. The Boatswain, who knows a thing or two about sailing, basically tells the Duke of Milan to keep his mouth shut and get out of the way: "Hence! What cares these / roarers for the name of King? To cabin! Silence! / Trouble us not" (1.1.16-18).
Oh, snap! The social and political hierarchy begins to break down here as the Boatswain points out that royal titles are meaningless in a life and death situation at sea. ("Roarers," by the way, means "waves" in this context but it's also a term used to describe a rioting crowd.)
At this point, we wonder for a moment if we'll have a mutiny on our hands that could turn into a Lord of the Flies scenario once everyone washes up on shore. (If Stefano, Caliban, and Trinculo had their way, Stefano would have usurped both Alonso's and Prospero's power, right?) When we think about it, this is basically what happened back on dry land in Milan when Prospero's brother snatched the dukedom from him.
Want to know more about Shakespeare's love of storms? Check out "Symbols" in King Lear and Macbeth.