How did Sigmund Freud impact 1920's America?
The first answer lays out some of Freud's ideas quite nicely but does not say anything about how he impacted America in the 1920s.
I would argue that most of Freud's impact came from his ideas about how central sexual desires are to human beings. He argued that many problems people have come from repressed sexual desires.
The 1920s in the United States were a time of loosening values and sexual values were loosened as well. Things that had been previously unthinkable (like women going out alone with men and like "petting" which came to be a term in those days) came to be much more accepted. You can argue that this reflects Freud's influence -- people thought that it was harmful to suppress sexual urges so much so they came to have a more open attitude towards sex.
Sigmund Freud led the way for what we now identify as psychotherapy. He was the fist man to create a therapeutic ideal for a mental patient. It was the first system of the first system of psychoanalytic psychology. Freud performed studies on neurosis, hysteria, self-analysis, and created the idea of the Id, Ego, and Super-Ego.
Sigmund Freud is considered to be one of the great thinkers of his day and the father of psychotherapy. Other areas that he impacted were his addressing psychosexual issues, dream interpretation, defense mechanisms, and psychosis. He created theories on personality development and the stages of development. He laid down the foundation for others such as young to follow and develop upon. Prior to Freud's studies and publications mental health existed as a unit in a asylum where patients were sedated and lived in filth clumped together in cages and dorms like animals. They did not receive therapy and some even believed their disorders to have been caused by the devil.