Here's what you need to know about John of Gaunt: he's Henry Bolingbroke's dad, he's really old-school, he's close pals with King Richard (at the play's beginning anyway), and his death is a major turning point in the play. Don't worry – we'll explain why all of this matters so much.
The best way to talk about Gaunt is to trace his development as a character. When Richard II opens, Gaunt seems like just another one the king's brown-nosers. Case in point: Gaunt knows that Richard is responsible for his brother's (Thomas of Woodstock's) death, but he and the other members of the nobility let Richard get away with it. When the Duchess of Gloucester asks Gaunt to avenge the murder, he basically says, "Sorry sweetie. God's the only one who can do anything, so go talk to him about it." Actually, Gaunt's speech is a lot more elegant and a lot loss smart-alecky. Check it out:
God's is the quarrel; for God's substitute,
His deputy anointed in His sight,
Hath caused his death: the which if wrongfully,
Let heaven revenge; for I may never lift
An angry arm against His minister. (1.3.2)
This is an important passage because it shows how Gaunt, like a lot of characters in the play, believes that Richard has basically been appointed by God to rule England, which means that he doesn't have to answer to anyone but God... even if he commits murder.
At the beginning of the play, Gaunt is so loyal to the king that he even suggests that his own son be banished for reporting Mowbray as a traitor. But then, when Richard accepts Gaunt's suggestion and exiles Henry Bolingbroke, Gaunt is completely devastated. To Richard he says, "You urged me as a judge but I had rather / You would have bid me argue like a father" (1.3.4). In a play that's chock full of situations that force characters to choose between political and family ties, Gaunt finally gets it: he realizes that being a good dad is more important than being loyal to the king, especially since the current one is so darn corrupt.
Unfortunately, Gaunt's big "aha" moment comes a little too late. Deprived of his banished son, sick and dying, he decides the time has come to offer the king real guidance. He reasons that "Though Richard my life's counsel would not hear, / My death's sad tale may yet undeaf his ear" (2.1.2). As a dying man, Gaunt tries to get Richard to listen to reason and advises him frankly on his deathbed (2.1). He tells Richard he's a bad king and he's making huge mistakes by mismanaging the kingdom. This is a pretty big deal, because Richard is all too willing to punish advisors who tell him stuff he doesn't want to hear. Gaunt's honesty shows us that he's brave, a trait he shares in common with his son. This is important, because ultimately, Richard's inability to listen to Gaunt's wise advice results in Gaunt's son taking the throne.
A lot of literary critics see Gaunt's death as a major turning point in the play. As Marjorie Garber points out, once Gaunt dies, "there is no going back" because it sets so many important events in motion. (Sort of the way Mercutio's death is a turning point in Romeo and Juliet.) After Gaunt dies, Richard seizes his land, which prompts Henry Bolingbroke to return to England in order to claim his birthright. In the process of demanding his dead father's land, Henry grabs the throne and changes England forever.
Aside from that, Gaunt's death seems symbolic because Gaunt was one of the last old-school members of the nobility who believed that Richard had a God-given right to rule. Go to "Themes: Power" if you want to think about this some more.
It just so happens that Gaunt delivers the play's most famous and stunning speech. Seems like we ought to take a look at it, don't you think? (If you're planning on reading this speech out loud, you'll want to take a very deep breath before you start.)
This royal throne of kings, this scepter'd isle,
This earth of majesty, this seat of Mars,
This other Eden, demi-paradise,
This fortress built by Nature for herself
Against infection and the hand of war,
This happy breed of men, this little world,
This precious stone set in the silver sea,
Which serves it in the office of a wall,
Or as a moat defensive to a house,
Against the envy of less happier lands,
This blessed plot, this earth, this realm, this England,
This nurse, this teeming womb of royal kings,
This land of such dear souls, this dear dear land,
Dear for her reputation through the world,
Is now leased out, I die pronouncing it,
Like to a tenement or pelting farm: (2.1.3)
Okay, let's break this down. The subject of this speech, of course, is England, which Gaunt refers to in a dizzying string of metaphors – "this royal throne of kings," "this scepter'd isle," "this earth of majesty," "this seat of Mars," "this other Eden," "this fortress," "this precious stone set in the silver sea," "this nurse," and so on. Did you notice that this description of England (21 lines long!) is basically just one long sentence?
We also want to point out that the speech is very, very patriotic, praising England as a kind of "other Eden," a "demi-paradise," and a "blessed plot" of earth that's protected by God and the sea that surrounds it like a "precious stone." Yet, despite Gaunt's praise, the speech gives way to something darker and less pleasant. By the time we near the end of the passage, Gaunt decides to come out and tell us, "Hey, guess what? That was the old England. Today, under Richard II's reign, England is "bound in with shame" because Richard has "leased out" royal lands.
Why does it matter what an old dying man has to say? Well, Gaunt gives voice to the play's sense that Richard has turned England into a fallen paradise. This is a big deal, because Shakespeare takes the idea and runs with it throughout the rest of this four-play history cycle. (We talk more about this is "Symbolism.")