Let's face it. Shakespeare rarely passes up an opportunity to make a joke about bestiality. (Just ask Titania, who falls in love with an "ass" in A Midsummer Night's Dream.) So, we're not really surprised when we meet Bourbon, a French nobleman who loves two things: warfare and his horse.
In fact, Bourbon is so crazy about his horse that he can't stop talking about him:
DAUPHIN I once writ a sonnet in his praise and began thus: 'Wonder of nature'— ORLÉANS I have heard a sonnet begin so to one's mistress. DAUPHIN Then did they imitate that which I composed to my courser, for my horse is my mistress. ORLÉANS Your mistress bears well. (3.7.40-47)
Say what?! Dauphin's horse is his "mistress" (a.k.a. girlfriend)? This is kind of weird, don't you think? What's going on here? Well, aside from poking fun at the Frenchman, Shakespeare is also making a reference to a character from an earlier play: Hotspur (from Henry IV Part 1). Remember him? He's the guy who used to get all hot and bothered just thinking about going into battle. So hot and bothered, in fact, that he liked hopping on his horse and going off to war better than sleeping with his wife (Henry IV Part 1, 2.3). Because of his love of battle, Hotspur was considered the epitome of chivalry (a word that comes from the French word for horse, "cheval"). When Dauphin implies here that he prefers his horse to women, Shakespeare basically turns Dauphin into an exaggerated version of Hotspur. In the process, he makes fun of traditional ideas about "chivalry" and cracks a dirty joke in the process.