For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
A rich heir named John Harmon gets murdered on his way to London to claim his inheritance. And that's not the only weird thing about the inheritance. Apparently, Harmon could only claim his fortune if he married Bella Wilfer, a girl he'd never met.
Oh well: none of it matters now because Harmon is dead and the money has been passed on to a pair of servants, Mr. and Mrs. Boffin. Meanwhile, a guy named Eugene Wrayburn falls in love with a working-class girl named Lizzie and uses every opportunity to insult her other suitor, Bradley Headstone.
Act II
Over time, Mr. Boffin gets greedy and paranoid about his newfound fortune. He hires a secretary named Mr. Rokesmith to handle his affairs. Then he takes in Bella Wilfer as a sort of adopted daughter and gives her all the money she wants, since he thinks a good chunk of the fortune is rightfully hers.
Meanwhile, a guy named Silas Wegg worms his way into Boffin's good graces and plans on blackmailing Boffin for most of his money when the time is right. It seems like Wegg has found an old copy of the Harmon will that leaves all of the money to the government and not Mr. Boffin. On top of all that, Eugene Wrayburn continues to court Lizzie, even though he knows full well that his rival Headstone is stalking him.
Act III
In the final act of Our Mutual Friend, we learn that Mr. Boffin's secretary, Mr. Rokesmith, is actually John Harmon. Harmon has been pretending to be dead because he wants to see whether Bella Wilfer would marry him for who he is and not just for his money. Bella passes his test with flying colors and the two get married.
The reappearance of Harmon basically blows Silas Wegg's blackmail scheme out of the water, and Wegg is sent packing. John and Bella have a baby daughter together and move to a fancy new house in London, prepared to start enjoying life as super-rich newlyweds. Meanwhile, Eugene Wrayburn nearly gets murdered by Bradley Headstone, but the attempted murder only brings Eugene and Lizzie closer together, and the two of them get married. Love conquers all!