The Tempest opens with a storm at sea. Travelling back from Tunis with Alonso, King of Naples, and Antonio who had previously usurped his brother Prospero's role as Duke of Milan, Gonzalo ('an honest old councillor') attempts to engage the Boatswain in an urgent debate about their prospects of survival:

Boatswain: Keep your cabins! You do assist the storm.
Gonzalo: Nay, good, be patient.
Boatswain: When the sea is. Hence! What cares these
roarers for the name of king? To cabin! Silence!
Trouble us not.
Gonzalo: Good, yet remember whom thou hast aboard.
Boatswain: None that I more love than myself. You are
a councillor. If you can command these elements to
silence, and work the peace of the present, we will not
hand a rope more. Use your authority. If you cannot,
give thanks you have lived so long… (1.1.14-25)

This immediately brings into question the nature of authority. The King and his court cannot control the storm. Unknown to the crew, however, Prospero does control it.
His daughter Miranda watches from the shore, and she has her suspicions:

Miranda: If by your art, my dearest father, you have
Put the wild waters in this roar, allay them.
The sky it seems would pour down stinking pitch,
But that the sea, mounting to th'welkin's cheek,
Dashes the fire out. O, I have suffered
With those that I saw suffer! A brave vessel,
Who had, no doubt, some noble creature in her,
Dashed all to pieces. O, the cry did knock
Against my very heart! Poor souls, they perished.
Had I been any god of power, I would
Have sunk the sea within the earth, or ere
It should the good ship so have swallowed and
The fraughting souls within her.
Prospero: Be collected.
No more amazement. Tell your piteous heart
There's no harm done.
Miranda: O, woe the day!
Prospero: No harm.
I have done nothing but in care of thee,
Of thee, my dear one, thee my daughter, who
Art ignorant of what thou art, naught knowing
Of whence I am, nor that I am more better
Than Prospero, master of a full poor cell,
And thy no greater father. (1.2.1-21)

This passage introduces Miranda as a sensitive and sympathetic figure, and also shows a lack of communication between father and daughter. Miranda finds her suspicions confirmed, and cries out; Prospero tries to calm her (note the repetitions 'of thee'). He then has to embark on an explanation of the 'back-plot'. His explanation is fairly tortuous, and he frequently checks to see if Miranda is following him. He then charms her to sleep, and gets a report on the storm from his spirit Ariel. Ariel wants freedom.

Prospero: The time 'twixt six and now
Must by us both be spent most preciously.
Ariel: Is there more toil? Since thou dost give me pains,
Let me remember thee what thou hast promised,
Which is not yet performed me.
Prospero: How now? Moody?
What is't thou can demand?
Ariel: My liberty.
Prospero: Before the time be out? No more. (1.2.240-246)

Prospero bullies Ariel (who has created the storm) into renewed submission.

Prospero: If thou more murmur'st, I will rend an oak,
And peg thee in his knotty entrails, till
Thou hast howled away twelve winters.
Ariel: Pardon, master.
I will be correspondent to command,
And do my spriting gently.
Prospero: Do so, and after two days
I will discharge thee.
Ariel: That's my noble master!
What shall I do? Say what! What shall I do? (1.2.294-300)

This is not the end of Prospero's difficulties, however. After Miranda and Ariel have been controlled, we are introduced to the recalcitrant Caliban, son of Sycorax.

Caliban: I must eat my dinner.
This island's mine, by Sycorax my mother,
Which thou tak'st from me. When thou cam'st first
Thou strok'st me, and made much of me; would'st give me
Water with berries in't; and teach me how
To name the bigger light, and how the less,
That burn by day and night: and then I loved thee
And show'd thee all the qualities o'th'isle,
The fresh springs, brine pits, barren place and fertile:
Curs'd be that I did so! All the charms
Of Sycorax, toads, beetles, bats, light on you!
For I am all the subjects that you have,
Which first was mine own King: and here you sty me
In this hard rock, whiles you do keep from me
The rest o'th'island. (1.2.332-346)

Caliban was himself usurped, by Prospero. He is not grateful.

Caliban: You taught me language; and my profit on't
Is, I know how to curse. The red plague rid you
For learning me your language! (1.2.365-367)

When Miranda first sees Ferdinand (son of Alonso, King of Naples) she is impressed.

Miranda: I might call him
A thing divine, for nothing natural
I ever saw so noble. (1.2.418-420)
Miranda: There's nothing ill can dwell in such a temple.
If the ill spirit have so fair a house
Good things will strive to dwell with't. (1.2.458-460)

This shows the Neo-Platonic notion that beauty equals virtue. Caliban repeats his error when he meets the servants Stephano and Trinculo, enslaving himself, and hoping they will overthrow Prospero. In Act 3 he gives this lovely speech.

Caliban: Be not afeard; the isle is full of noises,
Sounds and sweet airs that give delight, and hurt not
Sometimes a thousand twangling instruments
Will hum about mine ears; and sometimes voices,
That, if I had then wak'd after long sleep,
Will make me sleep again: and then, in dreaming,
The clouds methought would open, and show riches
Ready to drop upon me; that, when I wak'd,
I cried to dream again. (3.2.133-141)

When Caliban's futile rebellion interrupts the betrothal Masque for Ferdinand and Miranda, Prospero relates his theatrical illusion (within the play) to the larger world:

Our revels now are ended. These our actors,
As I foretold you, were all spirits, and
Are melted into air, into thin air:
And, like the baseless fabric of this vision,
The cloud-capped towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve,
And like this insubstantial pageant faded,
Leave not a wrack behind. We are such stuff
As dreams are made on; and our little life
Is rounded with a sleep. Sir, I am vexed;
Bear with my weakness; my old brain is troubled. (4.1.148-159)

Prospero shows signs of exhaustion here. He becomes unnecessarily angry about Caliban's plot, and not until Act 5, with everyone under his control, does he relax.

Ariel: if you now beheld them your affections
Would become tender.
Prospero: Dost thou think so, spirit?
Ariel: Mine would, sir, were I human.
Prospero: And mine shall. (5.1.17-20)

He is persuaded to sympathy by a spirit who has no human feelings.

Prospero: Yet with my nobler reason 'gainst my fury
Do I take part. The rarer action is
In virtue than in vengeance. (5.1.26-28)
He then bids farewell to magic:
Prospero: graves at my command
Have waked their sleepers, oped, and let 'em forth
By my so potent art. But this rough magic
I here abjure, (5.1.48-51)

Ferdinand gives his summation of events on discovering his father is alive.

Ferdinand: Though the seas threaten, they are merciful;
I have curs'd them without cause. (5.1.178-179)

Miranda, seeing the Court gathered, is impressed. Her father is not so sure.

Miranda: O, wonder!
How many goodly creatures are there here!
How beauteous mankind is! O brave new world,
That has such people in't.
Prospero: 'Tis new to thee. (5.1.181-184)