Iambic Pentameter, Prose
Reading any one of Shakespeare's plays can feel like reading a long poem and that's because they're written in a combination of verse (poetry) and prose (how we talk every day). (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.)
We break all of this down in the paragraphs that follow, but here's what you should remember about Shakespeare's plays. The nobility and other important figures tend to speak in "blank verse," which is a formal way to talk. The commoners, or "everyday Joes," tend to speak just like we do, in regular old prose.
However, in The Tempest, Shakespeare does something a little bit different. Prospero's slave, Caliban, speaks prose (especially when he's cursing at Prospero), but he also speaks a lot of verse, which makes sense since Prospero taught him to talk.
Blank Verse or, Unrhymed Iambic Pentameter (The nobles)
Let's talk about 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 (sounds like da DUM). "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
Let's try it out on these lines from The Tempest, where Gonzalo comforts the shipwreck survivors:
beSEECH you, SIR, be MErry. YOU have CAUSE,
so HAVE we ALL, of JOY; for OUR esCAPE.
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.
Prose (Commoners)
Like we said, not everyone in the play speaks in verse. "Everyday Joes," (with the exception of Caliban) don't usually talk in a special rhythm—they just talk. Check out the way the drunken Trinculo speaks when he bags on Caliban:
I shall laugh myself to death at this puppy-headed
monster. A most scurvy monster! I could find in my heart to
beat him— (2.2.7)
Yup. No fancy iambs there.