Here are some explanations to some of the allusions that appear in his texts and might not be widely understood today. These examples were put together by Heloise Senechal, from The Complete Works of Shakespeare (2008) published by the RSC and Macmillan.

From The Third Part of Henry the Sixth, Act 3, Scene 3

GLOUCESTER : I'll drown more sailors than the mermaid shall, 
I'll slay more gazers than the basilisk, 
I'll play the orator as well as Nestor, 
Deceive more slyly than Ulysses could, 
And, like a Sinon, take another Troy. 
I can add colours to the chameleon, 
Change shapes with Proteus for advantages, 
And set the murderous Machevil to school.

mermaid the siren of classical mythology was said to sing sweetly in order to draw sailors onto rocks where they would drown

basilisk mythical reptile whose gaze had the power to kil 

Nestor Greek leader who fought at Troy; famed for his wisdom and eloquence

Ulysses King of Ithaca and hero of Homer's Odyssey; noted for his cunning

Sinon in Virgil's Aeneid, the man who pretended to desert the Greeks and persuaded King Priam of Troy to admit the wooden horse into the city, as a result of which Troy was destroyed 

Proteus the sea god Neptune's herdsman, who had the ability to change shape at will

set … school teach Machiavelli a few things 

Machevil an intriguer, an unscrupulous schemer (from Niccolò Machievalli's The Prince, a sixteenth-century treatise perceived as advocating ruthless political cunning)

From Much Ado About Nothing, Act 2, Scene 1

Benedick energetically declares his loathing of Beatrice.

BENEDICK: … I would not marry her, were she endowed with all that Adam had left him before he transgressed. She would have made Hercules have turned spit, yea, and have cleft his club to make the fire too. Come, talk not of her, you shall find her the infernal Ate in good apparel. I would to God some scholar would conjure her…

all … transgressed i.e. the delights and dominion of the Garden of Eden before the Fall

Hercules … too in Greek legend the Queen of Lydia, Omphale, made Hercules dress as a woman and spin wool while she took over his 

club and lion's skin; turning a roasting spit was one of the lowliest domestic tasks

Ate Greek goddess of discord and vengeance 

conjure exorcize (the sending of evil spirits back to hell had to be carried out in Latin so a scholar would be required)

From The Tragedy of Richard the Third, Act 1, Scene 3

Richard addresses Elizabeth, wife to his brother Edward IV.

RICHARD: . . . Was not your husband 
In Margaret's battle at St Albans slain? 
Let me put in your minds, if you forget, 
What you have been ere this, and what you are: 
Withal, what I have been and what I am.

MARGARET: A murd'rous villain, and so still thou art. 
Poor Clarence did forsake his father, Warwick, 
Ay, and forswore himself – which Jesu pardon! –

husband … slain historically, Elizabeth's first husband, Sir John Grey, was killed fighting for the Lancastrians, although in 3 Henry VI (Act 3 scene 2) he is depicted as having fought for the Yorkists 

battle army

St Albans a town about twenty-five miles from London

ere this before now 

Clarence … Warwick Clarence and the Earl of Warwick deserted the Yorkists; Clarence married Warwick's daughter, but later returned to the Yorkist cause, events that are depicted in 3 Henry VI 

father father-in-law 

forswore himself broke his oath (of loyalty)

From Troilus and Cressida, Act 2, Scene 2

Troilus describes Paris' role in the cause of the Trojan war.

TROILUS: And for an old aunt whom the Greeks held captive, 
He brought a Grecian queen… 
Is she worth keeping? Why, she is a pearl 
Whose price hath launched above a thousand ships, 
And turned crowned kings to merchants.

for in retaliation for 

aunt i.e. Hesione, Ajax's mother and sister of Paris' father Priam; she was rescued from a sea monster by Hercules and given to the Greek Telamon;aunt plays on a slang sense of 'bawd, old prostitute'

Grecian queen i.e. Helen (may pun on 'quean', i.e. prostitute) price … ships echoes Faustus' line to Helen in Marlowe's Doctor Faustus ('Was this the face that launched a thousand ships?') 

above more than

From Measure for Measure, Act 1, Scene 3

CLAUDIO: Thus can the demigod Authority 
Make us pay down for our offence by weight 
The words of heaven; on whom it will, it will, 
On whom it will not, so.

down immediately

weight in full (literally, weighing rather than counting coins, in order to ensure their true value) 

The … heaven according to the Bible (specifically Romans 9:15, in which God says 'I will have mercy on him to whom I will show mercy').

Developed with Macmillan and the Shakespeare Birthplace Trust