Bushy, Bagot, and Green are King Richard's more or less interchangeable yes-men. Throughout the play, they brown-nose the king and, unlike John of Gaunt, they only tell him what he wants to hear. They also benefit from Richard's corrupt policies. For example, at one point, Henry Bolingbroke refers to them as the "caterpillars of the commonwealth" (2.3.11), which means they're a bunch of parasites who are devouring or destroying England. (We also want to point out that this "caterpillars of the commonwealth" comment picks up on the play's idea that England is like a lovely garden that's being trashed by Richard and his cronies. Go to "Symbolism" for more on this idea.)
At times it can pretty hard to distinguish between these three guys. Here's some info to help you keep them straight. Bushy tries to comfort the queen in Act 2, Scene 2 but ends up getting executed by Bolingbroke. Green delivers the news to the queen that Henry has invaded Britain with a giant army (2.2.2). He's eventually executed by Henry too. Bagot is the only one who doesn't get executed, because he runs away to Ireland as soon as he hears that Henry has invaded England (2.2.3).