Which figure of speech is used in this excerpt from Act 1 of Shakespeare's Romeo and Juliet?

ROMEO: O, she doth teach the torches to burn bright!
It seems she hangs upon the cheek of night
Like a rich jewel in an Ethiope's ear;
Beauty too rich for use, for earth too dear!

The answer is simile. 

The simile in these poetically stirring lines or Romeo and Juliet forms a comparison between Juliet ["she"] and "a rich jewel in an Ethiope's ear." In other words, Juliet is so stunning that her beauty is as noticeable as a jewel set against the darkness of an Ethiopian's ear.

A simile is a comparison between two dissimilar things or persons through the use of a specific word of comparison such as like or as, (than, or resembles are also used sometimes). The comparison must be between two essentially unlike things. In this way a simile, like all figures of speech helps readers to perceive things in vivid, new ways.

There are other figures of speech in this passage from Act I, however, simile is the only one fitting the choices given. For instance, the first line has two examples of alliteration with the /t/ "teach the torches" and /b/ of "burn bright"; also, "cheek of night" is personification.

This is personification but it is also good use of imagery.  Romeo is saying that Juliet’s beauty outshines torches.  Numerous torches would have been used to light the way especially at a party where the host would keep his house well lit.  The light given off would be enormous and Shakespeare has his leading lady dimming this light with her presence.  This comparison also tells us about Romeo’s intentions.  He is fixated on Juliet’s beauty and her silence.  He views her as a piece of jewelry that decorates the night for his amusement.  Some say he is young and naïve while others say he is wise to relish the moment.  In any case, it is a wonderful use of imagery to explain the viewpoint of a star-crossed lover.

What does "teach the torches to burn bright" suggest in Romeo and Juliet?

This quote is spoken by Romeo when he first sees Juliet at the Capulet's ball in act 1, scene 5. Overwhelmed by Juliet's beauty, he asks one of the Capulet's servants about her identity, but the servant claims not to know. Romeo, entranced by the mysterious girl, wonders at her beauty, which at this point is the only thing he knows about her. He says to the servant,

O, she doth teach the torches to burn bright!

In this line, Romeo is suggesting that Juliet's beauty is as radiant as light, in fact outshining the torches that illuminate the room. He says that she could teach those torches, already bright themselves, how to shine even brighter.

This is a recurring theme in how Romeo speaks of Juliet. He often likens her to lights or bright objects, such as in his famous speech, when upon seeing Juliet at her balcony he exclaims,

But, soft! what light through yonder window breaks?
It is the east, and Juliet is the sun.

In this line, he compares her to the sun, which is bright enough to cover the world in daylight. Romeo is so infatuated that he sees Juliet as a bright spot that consumes him and radiates through him.

By having Romeo compare Juliet to a torch in his first encounter with her, Shakespeare also is referencing some of Romeo's dialogue in act 1, scene 4, in which a lovesick Romeo keeps insisting to his friends that he wants no part in merrymaking or dancing but, being in a melancholy mood, would rather sit and watch. He tells them,

Give me a torch: I am not for this ambling,

and he later insists,

A torch for me: let wantons light of heart
Tickle the senseless rushes with their heels,
For I am proverb'd with a grandsire phrase;
I'll be a candle-holder, and look on.

It is notable, then, that upon seeing Juliet, Romeo abandons the torches he had insisted on carrying as a symbol of his lovesickness and now devotes his attention to this new beauty who is more radiant than the torches. She is his new light and idol.

Who says this quote in Romeo and Juliet, and what is its significance?

"O, she doth teach the torches to burn bright! / It seems she hangs upon the cheek of night / Like a rich jewel in an Ethiop’s ear"

In act 1 scene 5, Romeo first glimpses Juliet across the dance floor of Capulet's masquerade party. It is one of the most famous moments of love at first sight, or at least attraction at first sight. 

He begins with the aforementioned lines:

O, she doth teach the torches to burn bright! 
It seems she hangs upon the cheek of night 
Like a rich jewel in an Ethiop’s ear (Shakespeare 44-46).

Romeo begins a long line of figurative language talking of Juliet's beauty. Basically, he is coming up with creative ways to say she is hot. In fact she is so hot, she can teach torches how to burn and so beautiful that she shines against the night like a jewel in an exotic woman's earring. 

This is significant because Romeo had been moping around in depression from being scorned by Rosaline. He only decides to go to the party because Rosaline is going to be there. However, showing how young emotions can be excessive and ever changing, he goes from completely heart broken about Rosaline to head over heels for Juliet in a moment. Thus begins a roller coaster of emotional immaturity for the star cross'd Romeo and Juliet.

Is "O, she doth teach the torches to burn bright!" from Romeo and Juliet an example of personification?

This line from Shakespeare’s The Tragedy of Romeo and Juliet,

“O, she doth teach the torches to burn bright!”

happens in Act I, Scene V. Romeo and his band have entered the Capulet’s house and are looking at the people as they talk and dance. Suddenly, without any warning at all, Shakespeare has Romeo notice a beautiful girl dancing. He asks a servingman who the girl is. When the servingman says he doesn’t know, Romeo responds with the line “O, she doth teach the torches to burn bright!”

Is this line personification? Personification is the attribution of human qualities to something that is not human. In this case, we are talking about torches, which are obviously not human.

So, what human characteristic is Shakespeare giving the torches? The ability to learn. He is saying that Juliet can “teach” them to burn bright. Learning is a human act, not something a torch can do.

Shakespeare’s point here is that Juliet’s beauty is so great that she could be considered the source of torches’ bright light. She is the one that taught the torches to create that  beautiful light.