Short story written by J.D. Salinger and published in his collection Nine Stories.

Like Salinger's greatest novel The Catcher In the Rye, this story is about imagination and the simplicity of being a child.

The main character is Seymour Glass, an ancient soldier who's in vacation at the beach with his superficial girlfriend, Muriel. Muriel and especially her parents believe that he's having serious psychological problems since he came back from war: he supposedly did some strange things with the trees, insulted Muriel and broken her father's car.

During the story, Seymour meets a nice little girl at the beach, Sybil. He talks with her for a while, kid her a lot and they both enjoy it, using his imagination and acting like a kid with her. After that, when he goes back to his hotel room, he starts kidding a lady in the elevator the same way he did with Sybil, but obviously she rejects him, thinking he is a kind of maniac.

Right after that, he went in his room and shot himself.

If you're wondering about the title, here's the part that explains it:

(Setting: Seymour and Sybil are playing together in the sea)

"Miss Carpenter. Please. I know my business," the young man said. "You just keep your eyes open for any bananafish. This is a perfect day for bananafish."
"I don't see any," Sybil said.
"That's understandable. Their habits are very peculiar." He kept pushing the float. The water was not quite up to his chest. "They lead a very tragic life," he said. "You know what they do, Sybil?"
She shook her head.
"Well, they swim into a hole where there's a lot of bananas. They're very ordinary-looking fish when they swim in. But once they get in, they behave like pigs. Why, I've known some bananafish to swim into a banana hole and eat as many as seventy-eight bananas." He edged the float and its passenger a foot closer to the horizon. "Naturally, after that they're so fat they can't get out of the hole again. Can't fit through the door."
"Not too far out," Sybil said. "What happens to them?"
"What happens to who?"
"The bananafish"
"Oh, you mean after they eat so many bananas they can't get out of the banana hole?"
"Yes," said Sybil
"Well, I hate to tell you, Sybil. They die."
"Why?" asked Sybil.
"Well, they get banana fever. It's a terrible disease."