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
Siblings Cléante and Élise have a problem. Their father is a stingy old miser who keeps them trapped under his thumb by giving them only a very little money to live on. On top of that, both Cléante and Élise are in love and want to get married, but can't do so without their father's blessing. Harpagon wants Élise to marry a rich old man named Anselme, but Élise wants to marry the servant Valère. Cléante's situation seems to be even worse, since Harpagon himself wants to get married to Cléante's honey, a (poor) girl named Mariane.
Act II
After a failed attempt to borrow money, Cléante decides that he'll do everything in his power to undermine his Daddy's attempts to marry Mariane. A conniving woman named Frosine makes an unsuccessful play at getting money out of Harpagon through flattery, and the servant Valère makes all the other servants hate him by constantly trying to butter up Harpagon.
Act III
The majority of the action happens in this third act. Harpagon loses his mind when he finds out that someone has stolen the box of money he buried in his garden. A servant named Jacques accuses Valère of stealing it. Harpagon gets a police officer, demanding that Valère be hanged. Seigneur Anselme (the dude Mariane is supposed to marry) finally shows up, and it is soon revealed that he is actually the long lost father of both Valère and Mariane. Because he's a good Daddy and wants what's best for his children, Anselme basically buys Harpagon's consent for Élise to marry Valère and for Cléante to marry Mariane. The siblings get to marry their sweethearts and Harpagon gets to cuddle with his money. Everyone gets what they want.