It's stormy – you might even call it tempestuous – when we meet some characters on a boat, which is a decidedly bad place to be in a storm. We learn that the King of Naples and several of his attendants are on this boat, and that things are going so badly there's not much to do but pray. The boat splits in half and the people float off into the sea.
We cut to Prospero (on dry land), our main character, chatting with his daughter Miranda. We learn that Prospero was the source of the magic that caused the storm that sank this boat, and that he did it for good reason. However, he promises his sweet daughter that nobody was hurt in spite of all the fire, boat-splitting, and drowning that was clearly going on.
Prospero also tells Miranda that it's time she found out that she's a princess. Prospero says he used to be the Duke of Milan until his brother, Antonio, betrayed him and stole the dukedom (with the consent of the King) while Prospero was busy learning magic in his library (arguably, not really his job). After all the usurping (which is a great word for stealing something, especially positions of power), Prospero and the three-year-old Miranda were shuttled out to the ocean in a wreck of a boat. They ended up on this island, where the ex-Duke has raised his daughter for the last twelve years. However, a star is looking pretty lucky in the sky, so Prospero thinks the time is right for action and revenge.
We briefly meet his two servants. One is a delicate and airy spirit who was imprisoned in a tree by a witch for not being nasty enough (Ariel) and the other is the child of said witch and the Devil (Caliban). Guess who's Prospero's favorite.
Then we learn that mostly all the folks responsible for stealing Prospero's dukedom were on the sinking boat from the beginning of the play, and they're now scattered about the island. Alonso, the King who allowed the wicked Antonio to take Prospero's dukedom, fears he lost his son (the Prince) in the storm. The shipwrecked group – Alonso, Antonio, Alonso's brother Sebastian, and various lesser lords – set off to find Alonso's son, the lost Prince Ferdinand.
Meanwhile, the not-so-lost Prince is alive and convinced that his dad and everyone else from the boat are dead. His grieving is kind of soft-core, since he's already fallen in love with Prospero's daughter Miranda. Prospero accuses the shipwrecked Prince of being a traitor and puts Prince Ferdinand to the hard task of carrying wood. Ferdinand is happy to do this because his newfound love for Miranda makes work seem easy. (Aw.) On Ferdinand's second encounter with Miranda, he learns her name and promises to marry her. She also declares her love for him, though he is only the third man she has ever seen (the first two are her dad and Caliban, the son of the Devil).
Back with the search party looking for the Prince, everyone feels weary and assumes the guy is dead. A banquet appears in front of the shipwrecked group, set up by silent fairy spirits. Yes, this is weird, but the search party is hungry and wants to eat. Before they can dig in, a scary harpy monster shows up. This freaky harpy (a result of Prospero's magic) says that the sea took Prince Ferdinand in exchange for the wrong Alonso committed against Prospero many years ago. The harpy also points out that there are three traitors at the table. This traitor comment brings us to an important side-plot: Antonio and Sebastian, thinking Prince Ferdinand is dead, are plotting to murder Alonso so Sebastian can be king. This is messed up because Alonso is Sebastian's brother. Still, Antonio clearly has no conscience; he admits that he's never been bothered by stealing his brother Prospero's dukedom. So, back at the scene with the monster harpy, Alonso is disturbed and repents of his foul deed, but Sebastian and Antonio – not so much.
Switching back to the other group on the island, Prospero now accepts Ferdinand, saying that he was just testing the young man with all that hard labor. Since the Prince has worked for at least three hours carrying heavy wood, he has permission to marry Prospero's daughter.
There is also another side story going on: Caliban has been plotting with the King's drunken butler, Stefano, and jester, Trinculo (also drunk), to murder Prospero so they can rule the island. Caliban (very drunk) pledges to be Stefano's slave and kisses his feet way more than we are comfortable with. The drunken schemers are led off by Ariel playing music. Ariel leaves the group in a pool that smells like the lesser part of a horse to wait his master's orders. The trio eventually gets out of the muck pool and sets off to murder Prospero. However, Prospero sets hounds upon them, and the would-be-murderers run off. Eventually they come back and get made fun of for a bit, at which point Caliban repents and says he'll work to be in Prospero's good graces again.
That being dealt with, Prospero now goes to meet the shipwrecked King & Co. The harpy really shook up the King, so Alonso apologizes to Prospero and returns his dukedom. Prospero doesn't tell the King directly of Antonio and Sebastian's treachery, but neither of the traitors apologize or repent or even shuffle their feet. They don't learn a lesson. However, Prospero starts some banter about how he recently lost his daughter to the tempest too, commiserating with the King. Prospero changes the subject and asks if they'd like to see his cell (the place he lives). He pulls back the curtain covering his dwelling to reveal – you guessed it – two not-dead children very much in love.
Alonso rejoices to see his son, Ferdinand rejoices to show-off his new girl, and Miranda rejoices at seeing so many men – hence the line "O brave new world that has such people in it." Prospero promises to explain most of this eventually. Tonight he'll tell some of his life story and everyone will head back to Naples via ships in the morning. Prospero says he'll watch the kids get married, and then he'll retire to his dukedom in peace. He charges Ariel to make sure the ships get to Naples safely, and then frees him from the servant gig.
During the closing lines, Prospero speaks directly to the audience, and says they can free him from the island with their applause. It's like "clap if you believe in fairies" except it's actually the best playwright in Western history saying goodbye to writing plays. All around, it's pretty intense. Maybe you should read it some time.