Stephen is a power loom operator in Bounderby's factory. He married young, and his wife has since become a raging alcoholic. Stephen is in love with Rachael, another factory worker, but can't be with her because he can't get a divorce. After being framed for bank robbery, Stephen ends up dying from falling into a giant hole in the ground.
Most of the time in this novel, the factory workers are talked about as a large group. Dickens describes them living their difficult lives in Coketown, while Bounderby rants and raves about how they're all lazy and greedy. Louisa, on the other hand, is taught to see them as "Hands" – basically a mass of parts indistinguishable from the factory's actual machinery. Stephen counters all that by helping us find a face in the crowd. He's not just a "Hand," he is an individual and a human being. So much so, in fact, that he is willing to not join the union like the rest of the workers. In response, they completely shun and ignore him, making Stephen actually – and not just figuratively – entirely independent and alone.
Just before he wrote this novel, Dickens saw a real unionization and subsequent strike in the city of Preston. He wrote an essay about it (called "On Strike" and published in his Household Words magazine) and was pretty solidly on the side of the striking workers. Why do you think he makes Stephen, who seems to be the worker hero of this novel, refuse to unionize?
OK, so Stephen is obviously in a hellacious marriage. Actually, when you think about it, his personal life is probably the most soap-opera-ish of any character here. He married young, and his wife is now a borderline psychotic alcoholic. Stephen is so desperately in love with another woman that he almost purposely lets his wife die. Things go really bad for him when he is framed for bank robbery. On top of all that, he dies a pretty crazy slow death at the bottom of a hole in the ground.
Now we're going to be Captain Obvious here for a second. When there are several similar situations in a novel, there is usually intentional comparison going on. We see this when Stephen comes to ask Bounderby how to ditch his wife. Bounderby is all like "well, marriage is forever, for better or worse." Of course, then he immediately says that actually rich people with influence can get divorces. Stephen's only way out of his marriage is death (either his or his wife's). Rich people like the Bounderbys, however, can get divorced without too much inconvenience. Put all this together, and what do you get? It looks like the novel may just be using marriage to show us yet another disparity between the rich and the poor.
Many – probably most – of Dickens's novels feature what we'll call a Human Sacrifice. You can identify a sacrifice when someone pure, good, noble, and unbelievably innocent is needlessly killed off. The death comes too early, is usually protracted (you know, for a lot of meaningful death scenes and sad speeches), and can usually be blamed on The System in some loose way. In theory, the death is so moving that readers are spurred to action. In practice, these scenes proved so popular (especially when they featured children dying – for instance Nell in Little Nell and Jo the street sweeper in Bleak House) that fans would write Dickens letters asking for more of the same.
In this novel, the honor falls to Stephen, who is set up for it almost from the very beginning (that's why he is introduced as a man of "perfect integrity"). The death of this kind of character (here, he is killed by falling into a dangerous hole in the ground left unmarked by an uncaring factory owner) is good a way to make a point without having to make an argument. Dickens could have researched a bunch of facts about how badly factories treated their workers. Why does he avoid that and go for the Human Sacrifice instead?