About 65% of Measure for Measure is written in verse (poetry) and the rest in prose (how we talk every day).
We break all of this down in the paragraphs that follow, but here's what you should remember about Shakespeare's plays: generally speaking, the nobility (Angelo, the Duke, Isabella, and Claudio) tend to speak in "blank verse," which is a pretty formal way to talk. The commoners, or "Everyday Joes" (like Elbow, Pompey, and Mistress Overdone), tend to speak just like we do, in regular old prose.
(Note: The play Richard II is the one exception to this rule – it's the only Shakespeare play written entirely in verse – even the gardeners speak poetry.)
Here are some specific examples from Measure for Measure:
In Measure for Measure, the noble characters mostly speak in unrhymed iambic pentameter (also called "blank verse"). Don't let the fancy names intimidate you – it's really pretty simple once you get the hang of it. Let's start with a definition of iambic pentameter:
An "iamb" is an unaccented syllable followed by an accented one. "Penta" means "five," and "meter" refers to a regular rhythmic pattern. So "iambic pentameter" is a kind of rhythmic pattern that consist of five iambs per line. It's the most common rhythm in English poetry and sounds like five heartbeats:
da DUM, da DUM, da DUM, da DUM, da DUM.
Here's an example, where Isabella says she won't compromise her virtue to save Claudio:
we CANnot WEIGH our BROther WITH ourSELF.
great MEN may JEST with SAINTS; tis WIT in THEM. (2.2.16)
Every second syllable is accented (stressed) so this is classic iambic pentameter. Since the lines have no regular rhyme scheme we call it "unrhymed iambic pentameter," a.k.a. "blank verse."
Isabella's impassioned speeches are delivered in verse, which is befitting her social status and also her integrity.
Not everyone in the play speaks in verse. Ordinary folks, as we've said, don't talk in a special rhythm – they just talk. Check out the way Mistress Overdone communicates:
Thus, what with the war, what with the sweat, what
with the gallows and what with poverty, I am
Mistress Overdone is a bawdy and unruly figure, so it's fitting that she talks in plain old prose (especially given that she always seems to be going on about the sex industry).