What is the purpose of the balcony scene in Romeo and Juliet?

The "balcony scene" act 2, scene 2, is a pivotal moment in the play for several reasons. First of all, it is key to demonstrating the difference in approach between the two protagonists. While both are extremely young and impulsive, having fallen in love mere moments after meeting, Romeo's verbose language, which utilizes the trappings of courtly romantic poetry, betrays him as a rather impractical slave to love, happy to soliloquize about the beauty of his "bright angel." By contrast, Juliet's thoughts have taken a different turn, and she sets out the key conflict in the play:

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 is concerned with the fact that, although she loves Romeo, he is a member of an opposing family. In this scene, the two lovers discuss the difficulties which this presents to them, and we see how they approach the issue. Romeo's inclination is to belittle it, so swept away is he in his romantic daze—he is happy to dismiss the issue, stating "henceforth I never will be Romeo." Juliet is concerned that Romeo will be killed in the orchard "if any of my kinsmen find thee here," to which Romeo says his love will protect him.
The key intrigue in this scene is the question of which of the two attitudes will win out: Romeo's romanticism, or Juliet's pragmatic concern, despite her new passion for Romeo. Ultimately, at the end of the scene, they agree that Romeo should "send...word tomorrow" if he intends to truly marry Juliet, and she will follow him "throughout the world." By the end of the scene, then, the events of the play have been set in motion. We have seen the lovers' concerns raised and interrogated, to a certain extent, and have seen them decide to pursue love over common sense. This allows the audience to settle in, ready to observe how this will play out, and whether the central conflict—that of Montagues against Capulets—will be resolved as a result of this love pact, even if the lovers in a tragedy cannot be expected to survive.