Prose is most often used by Shakespeare to indicate characters who are not of noble birth.
What is prose? Prose is written or spoken in its ordinary form without any set meter or rhythm. This is different from poetry, which is the structure that most of Shakespeare's writing takes, because classical poetry requires rhythm/meter.
Who in the play uses prose? One character who most often uses prose is The Nurse. She, as a character, is supposed to be more common and somewhat lewd, even if she loves and cares for Juliet. For instance, when Juliet and her mother are discussing the possibility of her future engagement, the Nurse says "An honour? were not I thine only nurse, / I would say thou has't sucked wisdom from thy teat" (I.iii)
Mercutio also often speaks in prose, which lends itself to characterizing him as a combative character who cares less about propriety and is more of a hothead. This is apparent in such places as II.iv, when Mercutio describes Tybalt and later in the same scene when Mercutio harrasses the Nurse. What is important to note about Mercutio's and the Nurse's use of prose is that it serves to indicate a difference of context - they do use poetic meter when speaking to certain people for whom they have respect. This is emphasized by Mercutio's speech with Tybalt in III.i.
Some final characters who use prose, which again emphasizes their lower or more common natures, are the Musicians and Peter in IV.v, whose lines also help add an element of comedy to a dark play.