Arthur is King John's young nephew and, according to the play, he's the rightful (translation: legal) heir to the English throne. That makes little Arthur a SERIOUS threat to the current monarch, which is why King John sends a hit man to snuff him out.
If there's one lesson Shakespeare's plays offer little kids, it's this: don't give anyone the impression that you could be a contender for the throne. In fact, just don't tick off the king, period. Will Shakespeare's plays are littered with the corpses of children who have found themselves on the receiving end of a monarch's anger. Go talk to the Young Princes of Richard III if you don't believe us—they'll tell you all about what happens when you get in your uncle the king's way.
Fortunately for little Arthur, he's so charming and adorable that he manages to avoid getting murdered by Hubert, King John's hit man, who turns out not to be all that interested in burning the kid's eyes out with a hot poker. On the other hand, not long after escaping the hot poker, Arthur tries to run away and manages to fall off a tower and get himself killed. Pretty messed up, don't you think?
(By the way, nobody is really sure what happened to the historic Arthur. All we know is that he "disappeared" in 1203.)
So, why does Shakespeare have the kid jumping off a tower wall? It seems to us that Arthur's brief and sad life is a reminder of the randomness and cruelty of fate. From the get-go, this kid doesn't have a snowball's chance in hell of surviving. His mom hits the nail on the head when she says that the goddess of Fortune is definitely not on Arthur's side, even though he's got a birthright to the throne:
But thou art fair, and at thy birth, dear boy,
Nature and Fortune joined to make thee great.
But Fortune, O,
She is corrupted, changed, and won from thee;
Sh' adulterates hourly with thine Uncle John... (3.1.53-54; 56-58)
We sort of have to agree with Constance on this. Arthur should have been a king, but he spends his entire brief life being a pawn in other people's schemes: he gets totally used by his own mom and King Philip of France, who are both looking to further their own political agendas. Plus, his own uncle wants him dead.
Even though Arthur has a legal right to rule (he's the son of John's older brother Geoffrey, who would have been next in line to rule after King Richard I died), he's doesn't exactly have the strength or power to take the crown from King John. (Not by himself, anyway: that's why King Philip of France offers to back him up.) This issue is pretty much at the heart of the play's biggest theme: Power. Shakespeare is always interested in what gives a monarch a right to rule verses what makes a monarch fit to rule.
On the one hand, Arthur definitely doesn't seem ready to rule a country. After all, he's just a kid who still needs his mom to hold his hand and tell him what to do. Oh, wait: how different is that from John? Shakespeare definitely wants us to ask this question.
Anyway, just in case we forget how young and weak Arthur is, someone is always there to remind us. Throughout the play, Arthur is referred to as "little Prince," "young Arthur," and so on. And "little Arthur" doesn't seem to handle conflict very well. At one point, he even starts to cry because his mom is bickering with his grandma about who should be king. In fact, he tells us straight out that he doesn't want anything to do with the crown: "I would that I were low laid in my grave. / I am not worth this coil that's made for me," he cries (2.1.169-170).
So, Arthur might have a legal right to the throne, but that doesn't mean he'd be a good king.
That said, Arthur does show extreme courage and presence of mind in the one scene where he does take control of his life. In other words, the kid turns out to be pretty gutsy when it comes right down to it. While Hubert is preparing to burn out Arthur's eyes with hot pokers, Arthur doesn't let a moment go by without reminding Hubert of their past friendship, slowly but surely working on his emotions until finally Hubert decides he can't do it (4.1).
When Arthur ultimately dies, it comes as a result of him taking charge: he figures that he is doomed if he stays where he is, so he'd better make a run for it (4.3). The irony is that his death by falling from the wall of the castle comes at the exact moment when he stopped being in danger—when the English nobles are looking for him, and John has decided that he no longer wants him dead. All the same, we're left to wonder what kind of king Arthur might have grown up to be.