Henry, Earl of Richmond (a.k.a. "Richmond") is the guy who bumps Richard off the throne and becomes King Henry VII.
He's a curious character in the play. In one sense, when he arrives in Act 5, too much of the action has passed for him to truly be a major player. Fittingly, rather than add to the dramatic tension, Richmond presents himself as a deus ex machina, or an outside force that literally seems dropped from the gods to resolve the play on a high note. (Perhaps this explains Richmond's constant appeals and references to God.)
Richmond is at all times a noble and sweet creature; though he values peace, he does not fear achieving it through war. He treats his men as peers and is full of encouraging words rather than damning and threatening ones. Richmond is optimistic, humble, and sure that he and his men are fighting for all the right reasons. (Did you read the pep talk he gives his men before heading into battle at the end of Act 5, Scene 5? He even had us wanting to fight on his side.)
Richmond is something of a cardboard cut-out Disney hero, complete with dimpled chin. He is not fleshed out. He has no flaws; he even forgives his enemies. Really. After his victory at Bosworth Field he says, "Proclaim a pardon to the soldiers fled / That in submission will return to us" (5.8.3).
Because he's so perfect, it's hard to really love Richmond, but his lofty speeches give us the feeling that had we gotten to know him any better, we might have liked him well enough. In a sense, Richmond is only in the play to be a foil to Richard's wickedness – to be Richard's opposite. Richmond must be as much of a hero as Richard is a villain if he is to truly straighten out England.
One reason Shakespeare can't add any flaws to Richmond's character is the fact that he's the grandfather of Shakespeare's patron, Queen Elizabeth I. He's the guy whose reign ushers in the Tudor Dynasty and puts an end to the Wars of the Roses, which is a pretty big deal. So even though we don't get to see much of Richmond, his grand post-battle speech at the end of the play lets us know just how important he is. (Check out "What's Up With the Ending?" for more on this.)
If you want to learn more about the historical Richmond, who became King Henry VII, you can read about him on the BBC History website.