We get the big picture here: Henry is poor and there are some serious bugs around. We get to know our protagonist through his casual attitude. We also learn know that Henry is a romantic and philosophical fellow who spends a lot of time thinking about prostitutes and his next meal in equal measure. How is this guy going to get by without any resources?
Because this book doesn't move in a linear manner, we'll just tell you that Henry's main complication is getting a roof, some ladies, and a hot meal. This effort is not a proper conflict as such, but rather a series of conflicts—all of which he enjoys. Pretty much everything is a conflict for him, but his biggest charm (and most annoying habit) is that he faces everything with a kind of cheerful dismissal.
In a desperate and uncharacteristic move, Henry moves to Dijon to get a "real job" in a penitentiary/school. This is a big deal for someone who prides himself on not working. It's a bit of a "crisis" (in Henry's mind) because the place is just unbearable. At this point, we are asking ourselves: How's he getting out of this mess? We actually start to care that he doesn't get to be an over-sexed bum anymore! Man, he's a good narrator…
There's some serious doubt about whether Henry is a good friend throughout the novel. It's hard to tell what he wants out of friendships when all he seems to want is a hot meal or seconds on a prostitute. But toward the end, Henry does right by his friend Fillmore. Trapped (almost literally) in an ugly cluster of obligations, Fillmore has a nervous breakdown. Turns out he's not really mad, but just doesn't want to marry the abusive one-toothed peasant girl he's impregnated. Henry kicks into action and helps Fillmore ditch the lady. There's finally some resolution to someone's story—even if it's not Henry's.
Henry gets Fillmore on that train. Helping out wretched old Fillmore brings out a lot of new qualities in Henry, and we see him for the sort of decent guy that he is. Unable to see a grown man sob, Henry arranges to get Fillmore back to America. In the process, he gets a heap of money (not really meant to go to him) and begins to (once again) reflect on what he loves about Paris. Will he leave? Will he stay? The ending is actually kind of funny because it's not that different from the beginning. Though he momentarily considers returning to America, Henry re-realizes how good he's got it in the City of Lights.