Juliet goes through a wide range of emotions in Act 2 Scene 2. At first, she sighs and says, "ay me" showing that she is wistful, dreamy and obviously thinking about Romeo. Juliet gets annoyed with the whole situation as she is thinking aloud to herself about how unfortunately she is a Capulet and Romeo is a Montague.
"Deny thy father, and refuse thy name;
Or, if thou will not, be but sworn my love,
And I'll no longer be a Capulet."
This shows her desperation and frustration with the circumstances. It means that she is in love with Romeo, and not just list for him if she is willing to give up being a Capulet for him. You also get a sense of her longing for Romeo the way she talks about him.
Juliet then seems to seriously think about what hers and Romeo's names actually mean and plays with the idea of what makes a person.
She comes to the conclusion that Romeo is still the same man whatever his name and says,
"Romeo, doff thy name,
And for thy name, which is no part of thee,
Take all myself."
Suddenly, Romeo speaks and this is a big shock for Juliet as she was unaware there was anyone there. She is probably a bit embarrassed that someone had overheard her private thoughts, although she does not yet realise it is Romeo until he speaks again.
"My ears have yet not drunk a hundred words
Of that tongue's utterance, yet I know the sound.
Art thou not Romeo, and a Montague?"
When she realises that it is Romeo, she gets excited to see him. However, this soon turns to worry and she is fearful for him if any of her kinsmen find Romeo where he is not meant...