Verse and Prose
Henry VI, Part 2, like Shakespeare's other plays, is written in a combination of verse (poetry) and prose (how we talk every day). Usually, upper class characters like Henry, Warwick, or York speak in verse, while the commoners (Jack Cade and gang) use prose to get their thoughts across. It's one of the ways Shakespeare differentiates his characters: we can often tell their social statuses just from how they talk.
Verse
Shakespeare loved a little thing called "unrhymed iambic pentameter"—"blank verse," for short. Don't let the fancy names intimidate you —it's pretty simple once you get the hang of it.
An "iamb" is an unaccented syllable followed by an accented one (one of those pairs is called a "foot"). "Penta" means "five," and "meter" refers to a regular rhythmic pattern. So "iambic pentameter" is a kind of rhythmic pattern that consists of five iambs (or feet) per line. It's the most common rhythm in English poetry, and it sounds like five heartbeats: ba-DUM, ba-DUM, ba-DUM, ba-DUM, ba-DUM.
Let's try it out on these lines from Henry VI, Part 2:
The GAUdy, BLAbbing, AND reMORseful DAY
Is CREPT inTO the BOSom OF the SEA. (4.1.1-2)
Every second syllable is accented, and there are five of these feet, so this is classic iambic pentameter. Since these lines have no rhyme scheme ("day" and "sea" don't rhyme), we call it "unrhymed iambic pentameter," which is also known as "blank verse."
Now, blank verse is a pretty formal way to speak (have you ever met anyone who speaks that way?), so it's usually reserved for nobles and formal situations.
Prose
Now that we've gone over the verse, let's tackle the prose. Characters who don't get to speak in verse just talk. For example, take Cade's lines when he's talking to his men:
Be brave, then, for your captain is brave, and
vows reformation. There shall be in England seven
halfpenny loaves sold for a penny. The three-hooped
pot shall have ten hoops, and I will make it
felony to drink small beer. All the realm shall be in
common, and in Cheapside shall my palfrey go to
grass. And when I am king, as king I will be— (4.2.63-69)
See how there is no pattern to the lines? That's because he's just talking, without a meter to it. That's called prose. Cade is lower class, so he speaks in a more casual style than the nobles.