Prince Edward and his little brother the Duke of York (often referred to as "the princes") are the young sons of King Edward IV and Queen Elizabeth. Unfortunately for them, they're also the nephews of Richard III. Prince Edward (not to be confused with Lady Anne's first husband, Prince Edward) is the rightful heir to the throne, which makes him a major threat to his uncle Richard, who wants to be king.
These two little guys are witty and precocious – especially Edward. He knows it's a bad idea for him to stay at the Tower of London before his coronation – it's a prison, after all – and he also seems to know that Rivers and Gray were not his enemies, even though Richard claims they were: "God keep me from false friends," he says, "but they were none" (3.1.2).
Check out this exchange between Edward and Richard:
I fear no uncles dead.
GLOUCESTER (a.k.a. Richard)
Nor none that live, I hope.
An if they live, I hope I need not fear. (3.1.17)
In other words, little Edward isn't convinced that Richard (his living uncle and the guy who is supposed to be protecting him) isn't dangerous and he lets Richard know that he doesn't exactly trust him. Still, Prince Edward is just a little kid, so he's no match for his uncle.
In order to clear the path to the throne, Richard has the young princes murdered in Act 4, Scene 2. Up to this point in the play, Richard's actions have been evil and treacherous but somewhat tolerable given his charismatic ability to make the audience complicit in his behavior. When Richard kills off these two innocent kids, though, he goes way too far. (Tellingly, Buckingham, who has willingly gone along with Richard's evil plans, doesn't want any part of this wicked deed. Even the hired hit men feel bad about it.) So the princes' deaths speak volumes about Richard's sheer villainy. It's one thing to kill adult family members (we're talking about a family at civil war here); it's another thing to take out little kids.
Brain Snack: Historically speaking, it's unclear what actually happened to the princes. There's no actual proof that Richard III had them murdered. If you want to know more about this, check out the BBC's biography of little Edward.