In The Catcher in the Rye, "falling" or "The Fall" is an allusion to the "Fall of Mankind" in the Genesis "Garden of Eden" story. There, Adam and Eve partake of the Tree of the Knowledge of Good and Evil. Some people interpret this as having carnal relations. Regardless, their fall introduced humanity to sin, shame, guilt, and a state of "falleness" (separation from God, the supreme authority figure).
So it is for Holden, who is in a state of "falleness" from authority the entire book: he is separated from his parents, his school and teachers. Moreover, he is separated from his brother, sister, Jane, "friends," and even himself. Unlike Adam and Eve, who were exiled by God from paradise, Holden exiles himself after his fall from a corrupt society: he intentionally flunks out of school and runs away from home.
The title of the book refers to Holden wanting to be a "catcher in the rye" who rescues children from falling off a cliff. As such, he wants to be a teenage savior to children, preventing them from falling into the corrupt adult world as teenagers. He also wants to save them from death: Holden suffers from survivor's guilt over his brother's (Allie) death. In general, though, he wants to rescue kids from the adult worlds of materialism, sex, and hypocrisy (phoniness). In a Holden's perfect world, children would forever remain innocent, un-fallen and free of guilt, shame, and sin. In effect, he wants them and himself to stay like Allie--forever frozen in time.
Mr. Antollini himself tries to save Holden from falling. He says Holden is “in for a terrible fall.” His advice is:
"The mark of the immature man is that he wants to die nobly for a cause, while the mark of the mature man is that he wants to live humbly for one."
Mr. Antollini was the one who found James Castle (initials "J. C." for Jesus Christ--a martyr), who is Holden's romantic hero, after his fall. Antollini could not save James Castle from falling (jumping; suicide) from his window, but he did cover him up to prevent the rubberneckers from laughing at his dead body.
Antollini (critically believed to be the voice of Salinger) believes that Holden too is headed for the same fall as Castle (suicide). His advice quoted above suggests that he himself lives humbly for a cause, instead of dying for one. Holden's cause is demonstrated by the entire novel itself. The novel serves as a kind of "catcher in the rye" because it attempts to steer its readers away from falling from the dangerous cliffs of the corrupt adult world.