This is the famous "balcony scene" in Romeo and Juliet. The main irony in this scene is dramatic irony (when the audience knows something the characters do not know) and it comes at the very beginning of the scene. Juliet comes out to her balcony and is talking to the night air all about how attracted she is to Romeo. Romeo can hear her, and is himself responding under his breath, but she does not hear him, nor does she know he's there.
A key line in Juliet's speech is "wherefore art thou Romeo." Many students mistakenly interpret this to mean "Where are you Romeo?" But wherefore actually means why. What Juliet is essentially saying in this soliloquy is, "Why must you be a Montague? If you were the same person born under any other name, there wouldn't be a problem. Names don't really matter anyway, it isn't like your name defines you. Even if a rose was called something else, it would still look and smell like a rose..."
Then, Romeo reveals himself (and startles Juliet). She asked how he got in, he says he flew on the wings of love. She warns him that if he's seen he'll be killed, he replies that death would not be as bad as an angry look from her and then promises her he'll change his name, or do whatever else it takes to make her pledge undying love for him.
The rest of the dialogue is the two young lovers pledging in romantic metaphors how attracted to one another they are. At the end of the scene, Juliet promises to send Romeo word of a plan the next morning at 9. They exchange more words of love and devotion, and finally, Juliet goes inside and Romeo leaves, both hopeful that the morning will only bring good news.