The first quote from Romeo and Juliet that comes to anyone's mind is the famous "O Romeo! Romeo! Wherefore art thou, Romeo!" which has entered pop culture as a representation of the works of Shakespeare as a whole. For the tragedy of the young lovers, other, lesser-known quotes are more representative:
Tis torture, and not mercy. Heaven is here
Where Juliet lives, and every cat and dog
And little mouse, every unworthy thing,
Live here in heaven and may look on her,
But Romeo may not.
For Romeo, being apart from Juliet is "torture" because he understands how others, not separated as is he, are allowed to experience her presence. Romeo cannot, and he is miserable that even an "unworthy thing" such as a mouse may be with Juliet without consequence, when Romeo himself is denied on pain of death.
Another representative quote comes in the first scene, where Benvolio entreats Romeo to let his passion go:
Be rul'd by me: forget to think of her.
O, teach me how I should forget to think!
Again, Romeo's emotions are more powerful than his rational mind; he correctly explains (through melodramatic statement) that he cannot "forget" Juliet because his passion for her pervades every thought. He cannot simply "forget" to think about her because she is his reason for existing; he cannot learn to forget his love because the act alone would insult the truth of her beauty.
Juliet has some good lines as well; when Romeo kills Juliet's cousin Tybalt over a matter of honor and is banished, Juliet is torn between familial love and her greater passion for Romeo:
But with a rearward following Tybalt's death,
'Romeo is banished' -- to speak that word
Is father, mother, Tybalt, Romeo, Juliet,
All slain, all dead.
Juliet's love for Romeo is so great that she would have been less-affected by the deaths of her family members than by his banishment. She cannot reconcile her feelings, and so is emotionally distraught; Juliet would have been stricken by grief if her father or mother had died, but the banishment of Romeo is almost greater than his death, for it means that he lives but cannot be with her.