Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Exposition
Hank the Yank Wakes Up in Merry Olde England
After some storytelling somersaults setting up the situation—including changing narrators and otherwise shuffling the point of view around like a game of hot potato—Twain sets up the basic situation. The modern, practical Hank finds himself sent back in time thirteen centuries, to an era full of foolish things like magicians and men dressed in elaborate tin cans.
Rising Action
"We're gonna burn him at the stake! Oh wait, he's too smart for us!"
Hank soon has the primitive world wrapped around his little finger, thanks to his quick thinking and modern inventions. It's pretty one-sided—the poor knights of Arthur's court are way out of their league—which lets Twain point out how foolish and ridiculous the old Arthurian stories can be.
There's another, more depressing way of looking at the rising action, however. Most of it centers around Hank's efforts to civilize medieval society, bringing spoonfuls of rich, creamy social justice in the process. In the end, we realize that all of his efforts come to nothing: the rising action sets up the tragedy of the resolution.
Climax
No Rest for the Weary
For all the power he holds, Hank doesn't do much for himself. He does it all for England—for the poor and downtrodden he meets, and for the society as a whole, which he thinks he can improve. This changes when he marries Sandy and their baby gets sick. For the first time, Hank the Yank focuses on his family, taking Hello-Central to France to get better. In the month or so that he's gone, Arthur gets killed, the Church takes over, and all of his hard work gets tossed on the ash heap. Major bummer, thanks to Twain's cynical notion that no good deed goes unpunished.
Falling Action
War—What is it Good for?
Trapped in Merlin's cave with his followers, Hank makes a final stand worthy of John Wayne. With only fifty men, he takes down over 25,000 knights using weapons of modern warfare (a.k.a. guns). The bodies pile so high that they actually seal our heroes up in the cave.
And this is a good thing? Well, not really. Twain predicted the carnage of World War I pretty accurately here, and although we're rooting for Hank, he's probably just as sick as we are that it takes so much death for him to achieve victory.
Resolution
Get Out of Here
In the end, the book goes demonstrably from comedy to tragedy, as Hank falls into an enchanted sleep (thanks to Merlin) and wakes up back in the 19th century… without his wife, his baby, or the world he single-handedly changed. All his fighting and efforts come to nothing, and he seemingly dies of a broken heart in the book's last pages. It's a grim finale for an otherwise funny book.
You could also argue, though, that Hank had to go back because otherwise history would be all screwed up. In this interpretation, the tragedy emphasizes how nice it would be if Hank had really gone back in time… giving us modern civilization hundreds of years early and maybe fixing a lot of the world's problems in the bargain.