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.
Eliza Doolittle is a poor girl with a thick accent and no prospects. Henry Higgins and Colonel Pickering are gifted linguists. The three have a fateful encounter one night in Covent Garden, during which Higgins reveals his talents as a teacher.
OK, this is pretty standard. Shaw introduces us to the main characters, lets us know that Eliza has a problem, and that Henry has the skills to fix it.
The next day, Pickering and Higgins are working in Higgins's laboratory. Their conversation is interrupted by the entrance of Eliza. When the girl demands to be given lessons, Higgins bets Pickering he can pass her off as a duchess given six months. Pickering agrees.
Here we have the potential resolution. Higgins is now given the opportunity to fix Eliza's problems…but it's not all sunshine and rainbows. His motives for doing so aren't exactly clear.
Alfred Doolittle, Eliza's father, shows up and blackmails Higgins into giving him some money. Eliza is a quick study, but teaching her proper grammar and manners proves difficult. Freddy Eynsford Hill falls head over heels for her anyway. Higgins's mother warns him that he's only hurting Eliza by training her.
This is where things get, well, complicated. Eliza's father threatens to end the bet before it starts, teaching Eliza proves difficult, a romantic element is introduced, and a warning is issued. And rest assured, that warning isn't there for nothing.
After winning the bet, Higgins acts like he was completely bored by the whole process. He and Pickering proceed to talk about Eliza as if she hadn't even taken part in the plan. Eliza gets angry at Higgins and throws a slipper at him. Eliza decides to leave Higgins's home, and the two argue until Higgins loses his cool and nearly hits Eliza.
This is where things get a little unconventional. The winning of the bet, which you might expect to happen at the conclusion, is stuck smack dab in the middle of the play, and not much is even made of it. That's where the conflict in the scene comes from, and things wouldn't be nearly as exciting if Shaw didn't tinker with the formula.
Higgins shows up at his mother's house the next day looking for Eliza. She seems to have left in the middle of the night, and Higgins can't handle his daily affairs without her. He desperately wants to get her back, and even thinks about calling the police in to help search.
The old runaway plot. A classic. Eliza is lost, then found. It's a perfect way to build suspense and get us ready for the play's conclusion, but first…
It turns out that Eliza has been at Mrs. Higgins's apartment the whole time. She acts calm and collected, and gives Pickering most of the credit for her transformation, thus infuriating Higgins. When Eliza, surprised by the appearance of her father, howls as she used to before she was trained, Higgins declares victory. The two proceed to have a long argument.
This isn't really a denouement in the usual sense. You could lump the events listed about with Eliza and Higgins's final argument, but the two episodes are really distinct episodes. Still, a lot of the plot is tied together: Doolittle is reintroduced, Eliza is brought back into the picture, she goes on to explain how she feels, we see that her transformation isn't quite complete, and we're ready for the conclusive fight.
The argument, which focuses on Eliza's future, ends after Eliza threatens to sell Higgins's trade secrets to support herself. Higgins nearly strangles her, before deciding that Eliza has finally established herself as his equal. He invites her come back and live with him and Pickering again. Eliza declines and says goodbye for the last time. Higgins feels confident she'll come back anyway.
There isn't really anything about the conclusion that's conclusive, but we do get to see Higgins and Eliza talk about the many ways the play could have ended more conventionally: marriage, a total reconciliation and return to Higgins, a return to her father. Instead, we're left in the lurch. We don't know what's going to happen to her. She declares her independence from Higgins, but we don't know if it'll last. Though Shaw tells us what happens in his "sequel," well…go read "What's Up with the Ending?" to get the final verdict on that.