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
Act I is the beginning of the plot up until the point when the characters commit to some course of action. In The Maltese Falcon, Brigid O'Shaughnessy (under the false name of Miss Wonderly) hires Sam Spade and his partner, Miles Archer, to protect her from Thursby. But when Thursby Archer both wind up murdered the next day, Spade has to decide whether to turn Brigid over to the police or find a way to help her. He opts for the latter, and gets drawn into Brigid's mission to find the Maltese Falcon.
Act II
Act II is the point when the plot is furthest from its resolution. Brigid has run off and disappeared, seemingly into thin air, despite all of Spade's efforts to help her out (that's gratitude, for you). Spade is held at gunpoint and taken to Gutman, who finally tells Spade the full history behind the Maltese falcon. But before Spade can really question Gutman's story, he suddenly starts feeling dizzy. His drink has been drugged. Things are not looking too good for Spade so far.
Act III
The third act is when everything is resolve and wrapped up in a nice pretty bow (or, in this case, a not so pretty bow). It looks like all the cards are in Spade's favor: he has the Maltese falcon in his hands and is about to hand it over to Gutman in exchange for ten grand. But then Gutman realizes that the falcon is a fake, so the bad guys flee, leaving Spade empty-handed.
Brigid suggests that they run off together, but Spade throws her a curveball she isn't expecting: he accuses her of being Archer's murderer. When Brigid finally confesses to having committed the dark deed, Spade admits that even though he may harbor feelings for her, he has no choice but to turn her into the police. So in the end, the criminal does get sent off to jail, but we're still left with the feeling that the world still isn't really any safer.