The United States and its allies worried that if Vietnam fell to the communists, other countries in Southeast Asia would follow. This was known as "domino theory" -- that the spread of communism in Vietnam would result in other countries in the region falling like dominoes to the influence of communism.
President Dwight Eisenhower is the one who initiated the use of the term "domino theory." In a speech he gave in April, 1954, Eisenhower warned that if French Indochina fell to communism, it very likely could cause a domino effect in Southeast Asia. French Indochina is what the region of Vietnam was called during the days of French colonialism there. The movement against French colonial control was led by Vietnamese nationalist who also happened to be communists. The Vietamese nationalist / communist movement was led by HoChi Minh.
