As mentioned above NAM was founded in Belgrade in 1962. It was the idea of Indonesia's first president Nehru of India, Nasser of Egypt and Tito of Yugoslavia. The organization still exists today with 118 members.
Do you mean the Vietnam War? If so, you could say it started with World War II.
During WWII, the Vietminh, led by Ho Chi Minh, fought to free Vietnam from the Japanese. The Japanese had taken control of Vietnam from France in 1941. After the war, Ho believed that Vietnam would be given independence.
However, the French wanted to have Vietnam back as a colony. The United States agreed to allow this. At that point, the Vietminh started to fight France. After France pulled out, the US came in and eventually North Vietnam (as it was by then) and the South Vietnamese communists started to fight the US too.
During the Cold War a group of nations decided not to align themselves either with the US and its allies or the USSR. This group of countries was known as the NAM or the Non- Aligned Movement.
It was the conception of the leaders of Indonesia, India, Egypt and Yugoslavia and was founded in 1961. The aim of this organization was to follow a path that would decrease the tension that resulted from the Cold War and ensure that the World did not become two power blocs and lead to heightened aggression. Though the NAM still exists and has over 115 nations as its members, the relevance of the founding philosophy has reduced with the end of the Cold War.
I believe the acronym NAM in the question refers to Non Aligned Movement, which is perhaps the most popular topic in history with this acronym.
Non Aligned Movement (NAM) refers to an organization of countries and their activities that have adopted the policy of not being aligned to any one super powers in the world. This non-alignment was particularly relevant in the context of the intense rivalry that existed between USA and USSR during the Cold War days.
Members of NAM discuss and agree on policies for dealing with the major industrial nations. The leadership and headquarters of the formal organization rotate every three years. The NAM was formed following a conference attended by 29 Asian and African countries, held in Bandung, Indonesia, in 1955. The first summit meeting of heads of governments of NAM took place in Belgrade, Yugoslavia in 1961.
Following the end of the Cold war , NAM redefined its role as:
tackling the increasing polarization between North and South.
NAM has adopted the policies of giving support to democracy, human rights, economic globalization and free trade.