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
Egeus takes Hermia, Demetrius, and Lysander to Theseus's court. Theseus gives Hermia until his wedding to choose her husband. Lysander and Hermia tell Helena of their plans to run away; Helena tells Demetrius in hopes of earning his love back. He plans to follow Hermia and Lysander into the wood, and Helena plans to follow him. The workmen theater troupe enters the wood to practice Pyramus and Thisbe, starring Bottom. Oberon instructs Puck to smear the love potion on Titania's eyelids later that evening.
By this point in the story, all the dominoes are set up—it's just a matter of time for one to fall. Nothing crazy has happened yet to our three groups of characters, but each group is in place for the story's main conflicts to occur.
Act II
Let the trouble commence! Lysander and Hermia enter the wood en route out of Athens, Demetrius follows them, and Helena follows him. Oberon witnesses Helena chasing Demetrius and tells Puck to give the young man the potion as well. Puck gives Lysander the potion by accident, making the young man fall in love with Helena. Puck turns Bottom's head into a donkey head; Bottom then wakes up Titania by singing and she falls in love with him instantly. Puck smears some potion on Demetrius in an effort to right his earlier mistake; Demetrius falls in love with Helena as well. The boys fight over Helena's love, and Hermia chases Helena around. Pretty much, at this point, our characters' lives couldn't be more muddled up.
Of course, being king of the magical world, Oberon takes it upon himself to fix this mess. He orders Puck to 1) un-enchant Titania while Bottom's getting the full fairy treatment, 2) cast a shadow over the wood so that the lovers don't hurt one another, and 3) give Lysander the potion antidote, so he'll wake up back in love with Hermia. Everything is still all messed up, technically, but at least it's on the way to being sorted out. Night falls on the wood.
Act III
The next morning, Theseus and Hippolyta discover the two pairs of lovers sleeping in the wood and take them back to Athens to be married, since Demetrius still loves Helena and Lysander loves Hermia again. Bottom (sans donkey head) shakes himself off after the craziest night of his life and heads back to Athens where he finds his friends. After the weddings, the Mechanicals perform Pyramus and Thisbe for the court. Puck closes the play by promising that everyone will be happy and protected.