In Act 1, Scene 2, the Duchess of Gloucester makes a big speech to John of Gaunt about how King Edward III's seven sons are like "seven fair branches" on a family tree or seven vials of Edward's "sacred blood." Check it out:
Edward's seven sons, whereof thyself art one,
Were as seven vials of his sacred blood,
Or seven fair branches springing from one root:
Some of those seven are dried by nature's course,
Some of those branches by the Destinies cut;
But Thomas, my dear lord, my life, my Gloucester,
One vial full of Edward's sacred blood,
One flourishing branch of his most royal root,
Is crack'd, and all the precious liquor spilt,
Is hack'd down, and his summer leaves all faded,
By envy's hand and murder's bloody axe. (1.2.1)
Hmm, looks like someone has been spending a lot of time on Ancestry.com. But seriously, the idea is that if you're descended from King Edward III, your blood carries some of his sacred awesomeness with it. After all, it's the sacred blood of a king! That, says the Duchess of Gloucester, is why it's not cool that Richard II has had his uncle, the Duke of Gloucester, murdered. Gloucester was one of Edward II's seven sons, so when Richard had Gloucester killed, it was like he hacked down one of the "sacred branches" of Edward's family tree. (Psst. Check out this awesome genealogy map if you need to brush up on the royal family tree.)
By the way, we should point out that Shakespeare riffs off this family tree idea throughout the play. There are tons of references to plants, trees, and gardens, and people are always running around saying things like Henry has supplanted or uprooted Richard II as the king of England. By the way, Shakespeare is also having fun with the fact that Edward and his sons are part of the Royal House of Plantagenet. Get it?
Shakespeare is making a bigger point with all this sacred blood and seven branches talk, though. When too many people have royal blood, things get complicated. In a nutshell, Edward III had too many sons. Even though the rule is that the oldest son of the oldest son inherits the throne, it turns out that lots of Richard's cousins (like Bolingbroke) might actually take after granddad more than Richard himself.
In the play, Henry Bolingbroke justifies his return from banishment by arguing that if Richard is king by virtue of his blood, then he, Henry, deserves his inheritance too, for exactly the same reason. He's the son of a son of Edward III, just like Richard. However powerful Richard might think he is, and however many laws he might break, the one rule he can't break is the one that makes him king. If he goes against someone else who carries his blood, he risks giving his enemies a foundation to disinherit him too.