According to the theories of Sigmund Freud, psychological projection is a psychological defense mechanism whereby one "projects" one's own undesirable thoughts, motivations, desires, feelings, and so on onto someone else (usually another person, but psychological projection onto animals and inanimate objects also occurs). The principle of projection is well-established in psychology.
An illustration would be an individual who feels dislike for another person (let's say Bob), but whose unconscious mind will not allow them to become aware of this negative emotion. Instead of admitting to themselves that they feel dislike for Bob, they project their dislike onto him, so that the individual's conscious thought is not "I don't like Bob," but "Bob doesn't seem to like me."
Peter Gay describes it as "the operation of expelling feelings or wishes the individual finds wholly unacceptable too shameful, too obscene, too dangerous by attributing them to another." (A Life for Our Time, page 281)
The concept is anticipated in the Gospels and in the writings of Friedrich Nietzsche.
Psychological projection is the subject of Robert Bly's book A Little Book on the Human Shadow. "Shadow" refers to the projected material.