like a noose in this case).

The puns only make sense if you understand some old-fashioned slang. In Shakespeare's time, to 'carry coals' meant to be humiliated. That's where the conversation starts. 'Choler' is like fever (ever heard of cholera, the disease?), and a fever can be kind of like anger (or vice versa). And of course, nobody wants a hangman's collar around their neck.

In Act III Scene ii of Romeo and Juliet, Juliet's speech contains a triple pun on the sound that can be variously understood as "eye," "I," and "ay":

Hath Romeo slain himself? Say thou but ‘I,’
And that bare vowel ‘I’ shall poison more
Than the death-darting eye of cocatrice
I am not I, if there be such an ‘I’;
Or those eyes shut that make thee answer ‘I.’
If he be slain, say ‘I’; or if not, ‘no.’
Brief sounds determine of my weal or woe.

In other words, if the Nurse says "ay" (yes) to confirm Romeo has committed suicide, it will kill Juliet as surely as would the deadly eye of the legendary cocatrice. It will remove the reason for her existence ("I am not I") if this proposition is to be answered with an "ay" and Romeo's "eyes" are shut forever. Folding together the three meanings into one sound underlines how critical this fatal answer would be to Juliet's selfhood.

This wordplay serves a triple purpose: it is humorous in itself, it is a display of linguistic prowess that gives a good actor an opportunity to stand out, and it foregrounds Juliet's dilemma in this scene: her new husband has just killed one of her closest relatives (Tybalt) and has been banished for it, although at the point she delivers this speech, Juliet is still under the mistaken impression that the Nurse is referring to the death of Romeo himself.

In studying Shakespeare's puns and his frequent use of other "low" techniques, such as obscene jokes and allusions, we need to remember the nature of his audience. Many if not most of Shakespeare's plays were written for the public of his time and could be viewed at a cost the ordinary man could afford. As a prudent businessman, Shakespeare thus had to introduce elements that would appeal to viewers with the common man's sense of humor. The enormous range that this demand drew forth from him -- the fact that his writing has a wider appeal than merely to the classically educated gentlemen of his time -- is one reason why Shakespeare has been so consistently popular and has worn so well over the centuries.

A pun is a play on words that sound alike but have different meanings, or it could be one word that means two different things.

In Act 5, scene 3, the Chief Watchman finds the bodies of the County Paris, Juliet, and Romeo, and he calls for the Montagues and Capulets to be brought, to hear the news of their children. He says,

We see the ground whereon these woes do lie,
But the true ground of all these piteous woes
We cannot without circumstance descry. (5.3.194-196)

The pun here has to do with the word "ground" as it refers to both the earth on which the bodies lie, as well as the foundation of all the problems that might have caused these deaths. The watchman speaks about the ground on which the dead bodies now rest, but he also references the grounds of the discord between the two families. Further, he says that more investigation will be needed in order to totally understand the reasoning behind what has happened in the Capulets' burial vault.

A pun is a play on words.  There are puns throughout the play, and in the dramatic Act 5 they serve as comic relief.

Romeo’s man Balthasar greets him with a pun at the beginning of Act 5.  Romeo asks him if Juliet is well, because “nothing can be ill if she be well.”  He is basically saying that as long as Juliet is all right, everything is all right.  Unfortunately, Balthasar tells him that Juliet is dead.  He does so as a pun, however, which seems particularly insensitive even if it is kind of funy.

Then she is well, and nothing can be ill.
Her body sleeps in Capels’ monument,
And her immortal part with angels lives. (Act 5, Scene 1, p. 102)

The fact that she is well is incorrect, of course. It is a joke.  She is not really well, she’s dead!  Why Balthasar decides to make a joke at a time like this is questionable, but it certainly provides some comic relief for the audience when things are about to get very, very sad.  Part of the reason for the humor is a jolt to the audience, because Romeo is in his poetic lovesick state, and he is about to be pulled out.

Puns allow for jokes to be scattered in even the most serious scenes, as this one demonstrates.  Even though Romeo and Juliet is a tragic play, most of it is uproariously funny, and it is because of puns like this one.

A pun is a play on words, usually for humorous effect.  Shakespeare liked to use puns, often with sexual meaning.  However, puns can be based on just about anything. 

 Although puns are often made for humor’s sake, the person making them is not always laughing.  For example, Romeo makes a pun on the idea of being “in love.”

BEN:
In love?
ROM:
Out—
BEN:
Of love?(165)
ROM:
Out of her favour, where I am in love. (Act 1, Scene 1)

In this case, the pun is a play on the concept of being “in love” and Romeo is not really in a laughing mood.  It’s not really an incredibly sad scene though, because Shakespeare is still using the pun to inject some silliness.

Sometimes one character makes a pun off of the other.  Consider this exchange.

MER:
That dreamers often lie.
ROM:
In bed asleep, while they do dream things true.

The double meaning of “lie” is a pun, as in the fictional nature of dreams and being flat in bed.  In this case Mercutio begins the pun, and Romeo completes it.

The character of Mercutio uses puns a lot, often of a sexual nature.  However, his most famous pun is with his dying words.  Mercutio is such a punster that even with his last breath, he has to use a pun!

Ask for me to-morrow,
and you shall find me a grave man. (Act 3, Scene 1)

He will be a grave man, meaning serious, because he will be in his grave, because he will be dead!

Romeo and Juliet is one of the bawdiest plays the bard wrote, and one of the funniest.  This comes in large part from the puns.  The puns, largely coming from Mercuito and Nurse, were designed to entertain the cheaper seats in the theater, but would have brought a chuckle from even the nobility.  Even today, modern audiences can’t help but smile even in the most tragic scene when poor Mercutio is killed by Tybalt in a brawl and dies with a pun.

In act 1, scene 4 Romeo's friends want him to go with them to the Capulets' feast, but he is really not in the mood for entering into the party spirit. He'll go along with them, but without much enthusiasm; and he certainly won't participate in any dancing or general feasting. So he will simply stand around as a torchbearer, providing light for the ladies and gentlemen as they enjoy themselves:

Give me a torch: I am not for this ambling; 
Being but heavy, I will bear the light.

Romeo is engaging in a bit of punning here. He says that he is "heavy," meaning that he's feeling a bit down, but that he will "bear the light." Romeo is of course referring to the torch that he will carry during the festivities. The light is a heavy burden he would rather not be carrying. Figuratively speaking, Romeo is also carrying a torch for Rosaline, and it is his unrequited love for her that explains why Romeo is feeling so miserable.

When Romeo's friends are trying to talk him into going to the Capulets' party, Mercutio says that he wants to see Romeo dance.  Romeo replies, "You have dancing shoes / With nimble soles.  I have a soul of lead / So stakes me to the ground I cannot move" (1.4.14-16).  The pun is on the words soles and soul; Romeo says that Mercutio is carefree and ready to dance but that Romeo is not because he is so weighed down by his sadness. 

Later, when Tybalt and Mercutio are sword fighting and Mercutio is fatally wounded when Romeo comes between them, Romeo hopes that the wound is not so bad.  However, Mercutio knows that the injury is quite serious.  He says, "Ask for me tomorrow, and you shall find me a grave man" (3.1.101-102).  Here, the pun is on the word grave.  Grave has two meanings that apply here: serious and also the place where one is buried.  Both meanings work because it is a bad injury that will render Mercutio fairly serious (because he'll be dead) and because he'll be dead, he'll soon be put into a grave (i.e. he'll be a grave man because he'll be in a grave).

A pun, which is also called a paronomasia, is a word play that exploits the multiple meanings of words. There are a few characters in Romeo and Juliet that use many puns (possibly unintentionally), most notably the Nurse. Lesser known characters also use puns, however, and a great example is the serving men. The serving men are fairly unknown characters in Romeo and Juliet, but they have a brief scene that is one of the humorous exchanges in the play. The scene starts off with:

SERVING MAN 1: Where's Potpan, that he helps not to take away? He shift a trencher! He scape a trencher!
SERVING MAN 2: When good manners shall lie all in one or two men's hands, and they unwashed too, 'tis a foul thing. (I.v.1-4)

In this instance, Potpan's name becomes a pun. Potpan is the name of a missing serving man, but it is also an object that has not been taken away. This pun is a fairly silly joke (like most puns) that is often quickly glided over (or outright removed for the sake of shortening the play) in productions of Romeo and Juliet. 

As other’s have suggested, one important purpose of puns in Shakespeare's Romeo and Juliet is to lighten the mood so that the audience remains entertained throughout the play despite the tragic ending they know is coming.

Another important purpose of the puns is to have the audience see the skillful way in which Shakespeare plays with words. He shows off both his ability to craft an interesting story and his creativity with language and words. This also keeps the audience's interest as people listen for the different word games included in the dialog.

Shakespeare often used homonyms, or words that have the same sound but different meanings, to play with words. For example, Mercutio wants Romeo to dance at the Capulet’s party. However, Romeo’s heart is heavy because his beloved (not Juliet yet, but Rosaline) does not return his love. He declines to dance, saying:

"You have dancing shoes
With nimble soles. I have a soul of lead
So stakes me to the ground I cannot move."

Romeo is depressed because of his unrequited love for Rosaline. Shakespeare makes use of the similarity of “sole” and “soul” to explain why Romeo cannot dance. These kind of wordplays held audience member’s attention and also impressed Shakespeare’s chief patron, Queen Elizabeth, who was extremely well-educated and also a bit bawdy herself.

It is also important to bear in mind that audiences for Shakespeare’s plays in Elizabethan England did not sit quietly and listen to the words as modern audiences do. They often participated, applauding as the performance was going on when they liked the events or throwing things at the performers when they did not.

According to A Brief History of the Audience,

“Elizabethan audiences did not know what it meant to be quiet for a performance and would talk back to the actors.”

Thus, incorporating word play in the dialog is another device to keep the audience focused on the words, keep them amused, and elicit their attention to catch all the plays on words.

Tybalt accuses Mercutio of consorting—or associating—with Romeo. Shakespeare uses another play on homonyms that the Elizabethan audience would have understood. Consort could refer to Mercutio’s friendship with Romeo and it could also refer to a consort (group) of musical elements, as in the consort music that was popular at the time. Mercutio’s response to Tybalt illustrates the double entendre.

“Consort? What, does thou make us minstrels? As thou make minstrels of us, Look to hear nothing but discords.”

Shakespeare is showing the two possible uses of the word “consort” and also incorporating the word “cord” as “discord” to play on the theme of music. Mercutio is saying if you make minstrels of us, you will only hear poorly played cords and you will get discord—or a fight or disagreement—back.

Puns in Shakespeare's tragedies, such as Romeo and Juliet, serve the purpose of lightening the dark mood with humor to keep his audience drawn in and engaged.

In Romeo and Juliet, a few of the characters act as comic relief by making puns.

In the opening scene, Capulet's servants make several puns in their dialogue. Sampson uses the phrase "carry coals" to refer to humiliation. The act of carrying coals was the task that the chambermaid performed, the least significant household servant. Hence, carrying coals was seen as an insulting and demeaning task (Romeo and Juliet, eNotes). Therefore, when Sampson says to Gregory, "On my word, we'll not carry coals," he means to say that he will not allow themselves to be humiliated by the Montagues, making a pun out of the word "coals," which is referring both to literal coals and to humiliation. Gregory twists Sampson's words into a further pun in his reply, "No, for then we should be colliers." The term colliers, pronounced "coalers," refers not to chambermaids, but to "coal miners," forming another pun out of the word "coals." (eNotes).

The character Mercutio is also well known to make puns. In Act I, Scene 4, when Romeo says he feels uneasy about crashing the Capulet's feast due to a dream he had, Mercutio states, "That dreamers often lie," creating a pun with the word lie. Mercutio is using "lie" to refer both to sleepers lying down and to untruths. Hence, Romeo responds with, "In bed asleep, while they do dream things true," meaning, they lie down while being in "bed asleep," and they dream about things that are "true."

Earlier in this same scene, Romeo makes a pun of his own when he says,

Give me a torch: I am not for this ambling;
Being but heavy, I will bear the light.

Romeo is making a pun out of the word light, using it to refer both to literal light from the flame of a "torch" and to lightness of weight. Romeo is saying that since he feels sorrowful, or heavy hearted, he will carry the torch, thereby carrying the light, and becoming lighter in emotion.

A pun is a play on words in which two words are used that have the same sound but have different meanings. Many different puns can be found all throughout Romeo and Juliet.
One example of a pun can be found in the very first scene. When Sampson declares, "Gregory, upon my word, we'll not carry coals," meaning, we will not be humiliated by the Montagues, both Gregory and Sampson then make plays on the word "colliers" (I.i.1-2). To be a "collier" is to be a person who either digs for or sells coals (eNotes). However, said with a British accent, collier sounds very much like the word choler or collar. So when Gregory replies, "No, for then we should be colliers," Sampson turns "collier" into "choler," meaning angry, as we see in the line, "I mean, an we be in choler, we'll draw" (3). Greggory next turns the word into collar, which refers to a hangman's noose. Since this play on words can only be heard rather than seen, as the words are spelled differently, it can be difficult for a novice reader to catch. But basically collier, choler, and collar all sound the same and refer to coal workers, anger, and the hangman's noose respectively, making all three a play on words.
A second pun can be found in the Nurse's lines when we first meet her and Juliet in Act 1, Scene 3. When Lady Capulet asks Nurse Juliet's age, saying that she is not yet fourteen, Nurse replies by making a pun out of the word teen in the lines:

I'll lay fourteen of my teeth--
And yet, to my teen be it spoken, I have but four--
She is not fourteen. (15-17)

The word teen means sorrow, but can also be interpreted to refer to a teenager. Therefore, what Nurse is saying here is that she would bet "fourteen of her teeth, but to her sorrow she only has four teeth," making a pun out of the word teen to refer to both sorrow and Juliet as a teenager.