night! (Act 2, Scene 2)

This instant marriage proposal between two young people who either were or are supposed to be in relationships with others demonstrates the impulsivity of their decisions.  Juliet does exhibit some common sense by telling Romeo that a real proposal or acceptance would be worthless in this situation, but they still marry in secret soon after meeting.

In the beginning of Shakespeare's Romeo and Juliet, Romeo is very lovesick.  He's been chasing a girl named Rosaline, with whom he's pretty infatuated, and she has, apparently, been spurning his advances.  He is also pretty lusty.  When Romeo meets Juliet, however, he does seem very sincere.  In fact, before Tybalt kills Mercutio, Romeo was willing to let Tybalt kill him because he doesn't want to harm his wife Juliet's family.  Romeo's love for Juliet is in earnest, we can assume, or else he would not be willing to sacrifice his life in this exchange.

Juliet is somewhat more cautious when she and Romeo first meet.  She is flattered by his compliments, but she wants to proceed more slowly than he does in the beginning.  Although some may characterize her later actions as impulsive and unwise, I would classify Juliet as brave and strong.  At not quite fourteen years old, she is willing to drink the potion that will make her appear dead, have her body placed in her family's crypt, and await Romeo so they can run away together.  This would be pretty terrifying!  Although it doesn't work out, obviously, Juliet's bravery gave her and Romeo their best shot at being together. 

The first trait Juliet shows is that she is extremely sheltered. Consider where we first meet her: in her room (conversely, the first time we meet Romeo is on a street).

She is also inexperienced in romantic relationships. While her age is a good indication of that, we can also take the balcony scene as proof of this. Take this exchange:

Thou know'st the mask of night is on my face,
Else would a maiden blush bepaint my cheek
For that which thou hast heard me speak to-night
Fain would I dwell on form, fain, fain deny
What I have spoke: but farewell compliment!
Dost thou love me? I know thou wilt say 'Ay,
'And I will take thy word: yet if thou swear'st
Thou mayst prove false; at lovers' perjuries
Then say, Jove laughs. O gentle Romeo,
If thou dost love, pronounce it faithfully:
Or if thou think'st I am too quickly won,
I'll frown and be perverse an say thee nay,
So thou wilt woo...

She is so uncertain of how to act in this situation, and so she gives Romeo a few options of how she should act. Her insecurity in this shows us a lack of confidence from which he can infer her lack of experience in this sort of situation.

Romeo is impetuous, changeable, desire. He has some restraint, and some judgment—he doesn't start a fight in the Capulet party, or let himself get drawn into one—but mostly he's girl-crazy, and he finds his last love in Juliet.
Juliet wants to be a good and dutiful daughter—she tries to find ways to make the planned marriage to Paris work for her—but her ability to cut to the heart of things and see the potential for love in Romeo are what govern her.
Tybalt is fiery. He is anger and violence in action.
Mercutio is good with words. He likes to make jokes, and to smooth things over with words when he can. He has a naughty side.
For more on these characters, and the others, see that section of the enotes study guide on the play.

As far as why, they have these characteristics to set the play in motion, to make sure dramatic results happen, and to add tension.

One of Juliet's traits is stubbornness, which is especially seen in her desires to know her own mind and do her own will. We even see her stubbornness in the very first scene in which we meet her. Due to the time period, it is Juliet's responsibility to play the role of a dutiful daughter to both her parents, willing to do their bidding. In Act 1, Scene 3, we actually see that Juliet feels conflicted between her role as a daughter and her role to herself. When Juliet's mother asks Juliet to pay attention to Paris at the ball that night to see if she can like him, we especially see Juliet's budding stubbornness and her conflict between duty and self in her response:

I'll look to like, if looking liking move;
But no more deep will I endart mine eye
Than your consent gives strength to make it fly. (I.iii.101-03)

In these lines, Juliet is playing the role of a dutiful daughter by saying "yes" to her mother, but just barely. Her "yes" is canvased in an "if" clause. In other words, she's saying, "yes," she'll look to see if she can like him, but only if looking at someone can really determine whether or not you can like that person.
One of Romeo's traits is that he is a bit irrational and emotionally driven. We especially see his irrationalism in the very first scene. When he moans to Benvolio about his heartache over Rosaline's rejection, and Benvolio begs him to listen to his council and forget about Rosaline, Romeo's only response is, "O, teach me how I should forget to think!" (I.i.228). This is a very irrational and emotionally driven response. A wiser man than Romeo, like Benvolio, would understand that being rejected is not the end of the world, that allowing himself to continue to feel brokenhearted is unhealthy, and that it's time to move on. Hence we see that one of Romeo's traits is irrationalism as well as emotionalism.

What a fun question!  Well, obviously this is open to interpretation, but here is my opinion of the "best" character trait for each of the main characters:

Romeo: Passion (as evidenced by his every word and action and culminating in his death by poison)

Mercutio: Bawdy Humor (as evidenced by his efforts to break Romeo out of his "funk" with such stories as Queen Mab)

Juliet: Coy Intelligence (as evidenced by her flirtation with Romeo upon first meeting him and her innuendo in referring to men's body parts)

Tybalt: Abrasiveness (as evidenced by his desire to fight in ever scene he inhabits)

Nurse: Good-Natured Humor (as evidenced by her exchange with Juliet after returning with news from Romeo)

Paris: Chivalry (as evidenced by Paris' desire to lie in death near to Juliet)

Friar Laurence: Loyalty (as evidenced by his great lengths to both marry the young couple and desperate tries to help them live happy lives)

Romeo is quite emotional and rather impulsive.  For just about the entirety of the first act, Romeo is moping around and pining away as a result of his unrequited love for Rosaline.  His father says that he weeps every morning, "augmenting the fresh morning's dew," and shuts daylight out.  Even when his friends try to cheer him up and get him to crash the Capulets' party with them, he moans about how he doesn't feel like it.  Further, the day he marries Juliet in secret, Tybalt challenges him, and though Romeo initially refuses to fight, when Tybalt kills Mercutio, Romeo impulsively murders Tybalt in the street where everyone can see.  Had he been even a little thoughtful, he would not have done this (knowing the punishment is death).  Then, later, when he learns he's been banished, rather than be grateful, he cries some more and considers death a better alternative.  He seems, in many ways, immature as a result of his emotional volatility.  

Juliet, on the other hand, is often the more practical of the two, and she is very dedicated and loyal.  She is the one who suggests marriage, and tells Romeo how to proceed to get news to her (via the nurse she sends to him).  She braves her father's awful temper and terrible threats when he orders her to marry the County Paris, and she opts for a frightening plan to fake her own death in order to be with Romeo instead.  Even after Romeo rashly slays Tybalt, she remains loyal to him.  Her loyalty to Romeo -- despite his defects -- really drives the plot forward though it, sadly, also leads to the play's tragic end.