Like most of Shakespeare's plays. Henry V is written in a combination of verse (poetry) and prose (how we talk every day). For kicks, Shakespeare also writes some of the scenes in French. (Don't worry – he translates most of it for us.)
The Chorus and the nobility tend to speak in unrhymed iambic pentameter" (also called blank verse), which is a formal way to talk. 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.
Let's try it out on this line from Henry's famous battle cry:
once MORE unTO the BREACH dear FRIENDS, once MORE.
or CLOSE up THE wall WITH our ENGlish DEAD.
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.
Of course, not everyone speaks poetry in the play. A lot of the low-brow characters (like Mistress Quickly and Nim) talk in regular old prose. Check out this passage, where Mistress Quickly describes her last conversation with a dying Falstaff:
'How now, sir John!' quoth I 'what, man! be o' good
cheer.' So a' cried out 'God, God, God!' three or
four times. Now I, to comfort him, bid him a'
should not think of God; I hoped there was no need
to trouble himself with any such thoughts yet.
Like we said, Shakespeare writes a few scenes in French. (In Act 3, Scene 4, Catherine gets an English language tutorial. In part of Act 5, Scene 2, Henry tries to woo Catherine while her lady-in-waiting translates.) These scenes tend to be pretty hilarious because of the language barrier between the French and English characters. For example, when Alice teaches Catherine the words for various body parts: "la main" (hand), "les doigts" (fingers), "les ongles" (nails), "le bras" (arms), "le coude" (elbow). "le col" (neck), "le menton" (chin), "le pied" (foot), and "de cown" (gown), Catherine protests that "foot" and "gown" sound like the vulgar French words "foutre" and "con." Shakespeare gets a big kick out of making the French princess talk dirty so he has her repeat the naughty sounding words several times.