‘We are such stuff / As dreams are made on, and our little life / Is rounded with a sleep.’ These lines appear in Shakespeare’s The Tempest, in one of Prospero’s most famous speeches (‘Our revels now are ended’).
The Tempest is one of Shakespeare’s most enchanting and enchanted plays: a fantasy or ‘romance’ featuring a magician, the ‘monstrous’ offspring of a wicked witch, fairies, a lavish masque, drunken conspirators, young lovers, and much else.
Twelve years before the events of the play itself, a nobleman named Antonio overthrew his own brother, Prospero, from his position as Duke of Milan, because Antonio saw Prospero was more interested in tinkering about with magic than actually governing the city. Prospero and his daughter Miranda (who wasn’t quite three years old when this happened) were exiled from Milan and went to live on an enchanted island which is the setting for the play. Alonso, the King of Naples, supported Antonio in his usurpation of Prospero, as did Alonso’s brother, Sebastian.
The line ‘We are such stuff as dreams are made on’ appears in Act 4 of The Tempest. Prospero gives Ferdinand and Miranda’s union his blessing, telling Ferdinand that he must wait until the wedding-night to go to bed with Miranda. Ferdinand readily agrees.
They then watch the masque (a form of lavish court entertainment) that Prospero has arranged, celebrating their union: this spectacle is performed by spirits Prospero has conjured using his magic. The spirits embody the Roman deities Ceres (the goddess of the harvest), Iris (goddess of the rainbow), Juno (the female counterpart of Jupiter: so, ‘queen of the gods’, if you like), and Venus (goddess of love).
Once the masque is over, Prospero addresses his future son-in-law:
You do look, my son, in a moved sort,
As if you were dismay’d: be cheerful, sir.
Our revels now are ended. These our actors,
As I foretold you, were all spirits and
Are melted into air, into thin air:
Yes, this is the origin of our now familiar phrase ‘vanished into thin air’ to describe something that has disappeared as if 
it’s simply melted away into nothingness.
Why Prospero is attempting to reassure Ferdinand is a curious question in itself. Because Ferdinand only feels distress over the threat of insurrection from Caliban, Trinculo, and Stephano because Prospero was distressed by it, even though, with his magic (and with Ariel to help him), Caliban poses no serious threat.
This has led some critics of The Tempest to suggest that the masque involving Ceres etc. was an interpolation, added to the play for either theatrical reasons or to bulk out the existing play (The Tempest is not exactly a long play, even as it stands). If this is correct, then Prospero’s ‘Our revels now are ended’ speech was also a late addition, and Shakespeare must now bridge between the masque and Caliban’s attempt to overthrow Prospero and seize control of the island.
Prospero continues:
And, like the baseless fabric of this vision,
The cloud-capp’d towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Ye all which it inherit, shall dissolve
And, like this insubstantial pageant faded,
Leave not a rack behind. We are such stuff
As dreams are made on, and our little life
Is rounded with a sleep. Sir, I am vex’d;
Bear with my weakness; my, brain is troubled:
Be not disturb’d with my infirmity:
If you be pleased, retire into my cell
And there repose: a turn or two I’ll walk,
To still my beating mind.
‘We are such stuff as dreams are made on’ is often misquoted as ‘We are such stuff as dreams are made of’. But although people often correct those who alter ‘on’ to ‘of’, this is really a distinction without a difference, for Shakespeare is using the word ‘on’ to mean ‘of’ here, as Frank Kermode observes in his notes to “Tempest” (Arden Shakespeare).
What is less well-known is that Shakespeare’s ‘Our revels now are ended’ passage was drawing upon a now-forgotten play, Sir William Alexander’s Tragedie of Darius from 1603:
Let greatnesse of her glascie scepters vaunt;
Not sceptours, no, but reeds, soone brus’d soone brokē :
And let this worldlie pomp our wits inchant.
All fades, and scarcelie leaues behinde a token.
Those golden Pallaces, those gorgeous halles,
With fourniture superfluouslie faire :
Those statelie Courts, those sky-encountring walles
But this idea of palaces and courts being airy visions that melt to nothing when depicted on the stage was not exactly a new one even when Alexander used it, so as ever, Shakespeare’s ‘We are such stuff as dreams are made on’ was taking an existing trope and improving upon it.