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.
Exposition (Initial Situation)
Thrown Off Schedule
The country house is in an unusual state. Everything ran like clockwork before, but now that Serebryakov and wife #2 have arrived, everything's out of whack. This wrench in the works sets everyone up for the conflict to come.
Rising Action (Conflict, Complication)
Bizarre Love Triangles
Sonya is in love with Astrov, who ignores her, and everyone seems to be in love with Yelena, who ignores them. Also, Vanya has some serious hatred bubbling under the surface for Serebryakov. It's clear that something has to give because everyone is on the verge of ripping each other's heads off, which means that the play is getting complicated.
Climax (Crisis, Turning Point)
Shots in the Dark
Both Vanya and Astrov try their luck with Yelena and fail. Sonya tries to find out if Astrov has feelings for her, and it turns out he doesn't. To make matters worse, Serebryakov announces he wants to sell the country house, and Vanya tries to shoot him (but fails). There's really no turning back from trying to kill your brother-in-law, so that's what makes this the climax. Also, at this point, everyone has revealed their secret crushes, and everyone has been turned down, and it's hard to turn back from that, either.
Falling Action
Get Me Outta Here!
Yelena is ready to roll and leave the crazies behind. Vanya tries (and fails—are you starting to see a pattern?) to steal some morphine from Astrov. Basically everyone is abandoning ship. After the attempted shooting, everyone's just falling into place before the conclusion of the play.
Resolution (Denouement)
Right Back Where We Started
The Professor, Yelena, and the Doctor all leave, which means that Mariya, Sonya, and Vanya can all go back to their ordinary lives, paying bills and knitting stockings. Nothing has changed, which is an interesting kind of resolution. It's like Chekhov is saying that even after mega family conflicts, everyone just ignores their problems and carries on as usual.