Exposition (Initial Situation): France Gets All Up in King John's Face
Some people (like Arthur and King Philip of France) think Arthur should be the King of England, not John. Problem is, Arthur's just a kid and can't do anything about it, so King Philip challenges King John on the kid's behalf. John's not having any of that: he declares war on France.
Rising Action (Conflict, Complication): England and France Throw Down... Share Some Wedding Cake... Then Throw Down Again
When England and France go to war, it's pretty obvious we're at our "conflict" stage. It's a war, for crying out loud. This goes on until someone gets the brilliant idea that John's niece should marry the King of France's son, creating a convenient little political alliance between the two countries. (Arthur who?)
So John's reign as King seems safe... for about two seconds. That's when the Pope, who's got a beef with King John, sends a message saying that King Philip better stop being friends with John... or else. Philip doesn't want the Pope to be mad at him, so he does what he's told, and France and England throw down... AGAIN.
Climax (Crisis, Turning Point): King John Orders Arthur's Execution
It doesn't turn out so well for France or for little Arthur this time around. The French forces are defeated, and young Arthur is taken prisoner. King John orders Hubert to burn his eyes out (translation: kill him, but take the long route to get there), but Hubert can't bring himself to do it. He tells Arthur to lie low while he goes and deals with the king.
Falling Action: The English Nobles Rebel; Arthur Dies; France Invades England
John's plan of murdering Arthur backfires when it turns the English nobles are against him. Then, just when he seems to catch a break by learning that Arthur actually isn't dead… it turns out that Arthur actually is dead, which ensures that the nobles are dead-set against him. Oh yeah, and did we mention that the French are invading?
King John is so flabbergasted by all this that he turns command of his military forces over to the Bastard. Kings who are on top of their game aren't supposed to do this sort of thing.
Resolution (Denouement): King John Croaks—Peace Out
The war between King John's forces and those of Louis and the rebel English lords ends up being a bust all around. People buy the farm left and right; Louis's supply-ships get sunk on a sandbank; the Bastard's men get drowned by the tide. King John withdraws from the battle, feeling sick, and ends up getting poisoned by a monk.
Even though Prince Henry is around to take up his father's crown, things look dicey for him until it turns out that Louis has abandoned his invasion and is heading home for France. The official story is that Pandolf has brokered peace, but the real reason probably has more to do with the fact that all of Louis's English allies abandoned him once they learned he was planning to betray and kill them.
All in all, a downer of a finish—though the Bastard rounds things off with one last patriotic speech before applause and bows.