The chief concern that Juliet displays in this famous balcony scene involves her name and the way in which names hold such dominion over her life and that of her beloved. This is because, as both we and she knows, it is the name that Romeo bears and that she bears that are keeping them apart, as the feud that rages between their two rival houses means that any hope of a marriage between them that is recognised and accepted by their parents is an impossibility. Consider what Juliet says in this scene:
What's in a name? That which we call a rose
By any other word would smell as sweet.
So Romeo would, were he not Romeo called,
Retain that dear perfection which he owes
Without that title.
Juliet shows her ironic awareness of the fact that Romeo, were his name not Romeo Montague, would be an excellent suitor and husbnad for her. It is only his name that makes their happy union and marriage together impossible. Even a rose if it were called something else would "smell as sweet," and so Romeo, if he had a different name, would likewise be just as handsome and pleasing.
Juliet is concerned about his name, Montague. The Montagues are her family's enemy. She states only his name is her enemy, not the person. She even declares that if he cannot give up his name, she will give up hers.
Juliet is also concerned about Romeo's love for her. Keep in mind Romeo just fell out of love with Rosaline the day before, and Juliet is worried about his faithfullness. When Romeo compares his love for Juliet as the envious moon, Juliet is quick to tell him not to swear on it because the moon is constantly changing. She does not want Romeo's love for her to be changed like the moon changes.
Juliet is lastly worried about the new couple's wedding plans. She wants to marry him quickly, because of their family's hatred towards eachother. Romeo and Juliet both declare they would rather commit suicide than be seperated, and you cannot seperate a married couple. Juliet wants to be Romeo's wife so quickly, that she proposes to him, and expects the plans to be set by the next day!