The period of time during the 1920s when there was a fear of communism growing in the united states was known as the what scare.

  The RED SCARE

Explanation:

What historians refer to as the First Red Scare occurred from 1919 to 1921, following the end of World War 1 -- but more so following the Bolsvhevik Revolution which brought communism to power in Russia.   The Bolsheviks (meaning "the Majority") were the communist faction that led a successful overthrow of the regime of the tsar in Russia in 1917.   They weren't a "majority" in Russia, but they were the dominant group within the Russian communist movement.  Civil war in Russia followed during the next years, from 1917 into the early 1920s, ultimately leading to the establishment of the Union of Soviet Socialist Republics (USSR) in 1922.  There was fear in the United States (as there was elsewhere in the world) that communism would begin to spread further, beyond Russia.

The more common reference to "The Red Scare" usually refers to what historically was the Second Red Scare, from the late 1940s to late 1950s in the United States.  Following World War 2, as the Cold War developed and the Soviet Union was gathering allies, there was even greater fear -- and fear-mongering -- in the United States about the threat of communism.   The Second Red Scare was when  The House Un-American Activities Committee (HUAC) was created and when Senator Joseph McCarthy began a campaign of accusations against suspected communists in various sectors of American life.