We are introduced to Charlie Gordon, a man with an IQ of 68 who wants to be smarter. He is currently in a position to be chosen for an experimental surgery to do just that.
Climax
Charlie is tested in various ways to determine if he's good for the surgery or not, most notably racing to do mazes against a mouse who already underwent it, Algernon. He is decided to be a good subject and they give him the surgery.
Falling Action
Charlie grows very smart very fast and befriends Algernon, they make an impact in the science community and demonstrate at a convention in Chicago.
Resolution
As was feared by the doctors, Algernon regresses at an alarming rate and dies, having lost all superior intelligence.  Charlie, spurred by this and the threat of his own regression, races to find a cure.
Charlie continues to regress and finds that there's nothing he can do to stop it, yet continues to fight it while he can.
Charlie regresses fully, and leaves where he was, his last request to those he knew being for them to leave flowers on Algernon's grave, as he used to himself.