Jimmy Stewart was nominated for best actor, and Josephine Hull won the Oscar for best supporting actress for her portrayal of Veta.

Elwood P. Dowd (Jimmy Stewart) is a middle aged man whose best friend is Harvey, a 6 foot tall rabbit that only Elwood can see.  Elwood's sister and niece ,Veta and Myrtle Mae, want him committed to an insane asylum.  They think Elwood's odd behavior will make it impossible for Myrtle to find a husband.  Initially, Veta gets locked up instead.  Her lawyer gets her out, and the asylum staff go in search of Elwood.

When he finds Elwood, Dr Chumley (head of the asylum) also comes to believe in Harvey.  Elwood explains to Chumley that his mother had told him that to get by in life, you have to be either smart or pleasant.  After years of being smart, he decided to be pleasant, and it is much better.  Elwood's only goal in life now is to make the people around him happy.

They get Elwood back to the asylum, and Dr Sanders prepares to administer a treatment that will make Elwood "normal".  The driver who brought them tells Veta that it is too bad Elwood will be treated.  He says when he drives them out to the asylum, they are happy and friendly.  When he drives them back they are angry and mean.  Veta regrets bringing Elwood, and stops the treatment.  Elwood matches up Dr Sanders with Nurse Kelly.  He matches up Myrtle Mae with the tough guy orderly from the asylum.  Dr Chumley asks if Harvey can stay with him, and Elwood says whatever Harvey wants.

As they all head home, we see the gate open and close on its own as Harvey follows Elwood, indicating that Harvey is real.  

Thanks Joseph C!