You can see a clear illustration of the differences between Shakespeare's prose-speaking characters and his verse-speaking ones in the opening scene of the play. Flavius, who is a patrician-class character, comes into a street which is populated by—as the stage directions tell us—"Commoners." The audience, however, is not given these stage directions. As such, they are given clues in what the characters say as well as how they say it in order to understand what is going on in this scene, who is speaking to whom, and what kind of people they are.
Flavius speaks in verse. His verse also doesn't rhyme. Shakespeare, for the most part, constrained rhyming verse to certain positions. For example, at the end of a scene to give a sense that a conclusion has been reached. However, if you sound out what Flavius says, he is speaking in regular meter (an iambic pentameter common to all of Shakespeare's works). This meter indicates that he is a man of class and authority.
Immediately, the audience would have understood that this makes him different from the commoners that he is speaking to. They speak in prose, with no defined rhythm or meter. This indicates that they do not have the authority Flavius has: he is superior to them, according to the class divisions of Rome.
Shakespeare uses this technique throughout all of his plays. His major characters generally speak in verse while servants and lower-class characters will speak in prose. It is intended as a cue for the audience.