O Romeo, Romeo! wherefore art thou Romeo?
Deny thy father and refuse thy name!
Or, if thou wilt not, be but sworn my love,
And I'll no longer be a Capulet.
Juliet has fallen in love with Romeo at the Capulet party, but only realised as he exited that he was in fact a Montague. The Montagues and Capulets have a long-held feud, and Juliet is debating whether Romeo's name (i.e. Montague) makes any difference to Romeo himself.
Her first, famous line, asks "why are you Romeo?" (not "where are you, Romeo?") - and then asks him to forget his father and his name. Or, she says, she'll forget her name and no longer be a Capulet. She just wants to be with him.
'Tis but thy name that is my enemy.
Thou art thyself, though not a Montague.
What's Montague? it is nor hand, nor foot,
Nor arm, nor face, nor any other part
Belonging to a man. O, be some other name!
What's in a name? That which we call a rose
By any other name would smell as sweet.
So Romeo would, were he not Romeo call'd,
Retain that dear perfection which he owes
Without that title. Romeo, doff thy name;
And for that name, which is no part of thee,
Take all myself.
It's only ("Tis but") his name that is her enemy. And what is in a name? If you called a "rose" something else, it would still smell like a rose. And so, Romeo, even if he wasn't Romeo Montague, would be "perfect". Romeo, she asks, "doff" (take off) your name, and in return, take Juliet.
Quite simply, because of Juliet's love for Romeo by Act 2, Scene 2, she no longer sees the importance of either her last name ("Capulet") or Romeo's last name ("Montague") and, therefore, wishes to negate (or at least disregard) the age-old feud between the Montague and the Capulet families. First, Juliet asks, "Wherefore art thou Romeo? Deny thy father and refuse thy name! Or, if thou wilt not, be but sworn my love, And I'll no longer be a Capulet." In other words, Juliet asks why Romeo is even called "Romeo." She asks him to renounce his name and his father (in doing so). If Romeo won't do that, then Juliet asks for Romeo to say he loves her and Juliet will deny her name. Next, Juliet admits that "'Tis but thy name that is my enemy. / Thou art thyself, though not a Montague. / What's Montague? It is nor . . . any other part / Belonging to a man." In other words, Juliet says that it is only Romeo's last name (Montague) that is her enemy. A last name, therefore, has nothing to do with any body part that a person possesses. Further, Juliet says, "What's in a name? That which we call a rose / By any other name would smell as sweet. / So Romeo would, were he not Romeo called." In other words, Juliet suggests that if we grew up calling a rose a tulip, for example, that doesn't change the fact that it has a fragrance that only a rose can give. Likewise, Romeo would be just as handsome and loveable if his name were something else, like Eugene. Finally, Juliet says, "Romeo, doff thy name; / And for that name, which is no part of thee, / Take all myself." In other words, Juliet asks Romeo to disregard his connection to the Montague family and take her as a replacement. The brutal feud of many generations between the Montague family and the Capulet family is lost among these two lovers. They see no use or reason for the feud any longer, certainly not a reason to cease their love. In fact, perhaps Juliet's "What's in a name" speech could be said to contain the original idea behind the "Make Love, Not War" slogan.