Have you ever manipulated a situation to get your way? Have you ever seen someone's vulnerability, or someone who wanted you on his side so much, he was willing to make concessions? Hopefully, the last time you performed this tight-rope walk was when you were a child.
Non aligned nations were the leaders who figured out they didn't need to be pro Soviet or pro US. It was to their advantage to stay neutral and watch the superpowers' desperate acts: almost go to war, give aid, fund dams, etc.
Tito of Yugoslavia apparently figured this out. Under the hegemony of the Soviets, he constantly fought for his own autonomy which caused a separation and ultimately Yugoslavia was kicked out of the Soviet organization Cominform, and a call was made to isolate the country. After a disastrous harvest in 1949, Tito had to bow to the west for $150 million in aid. The Soviets considered invasion, but the US 6th fleet rested off the coast of Yugoslavia. Tito had played the US against the USSR and he was still standing.
In April of 1955, Tito, India's Nehru and Zhou Enlai of China hosted a non-alignment conference in Indonesia. They encouraged nations to retain their autonomy by encouraging neutrality in the Cold War.
The first great Cold War crisis of the Middle East happens soon after. Colonel Nassar, after deposing King Farouk of Egypt, asked the US to fund the Aswan Dam. They agreed. Nassar then buys arms from Czechoslovakia. The US cancels funding; the USSR jumps into fund the dam. After playing the US and USSr against each other, Colonel Nassar nationalizes the Suez Canal. France, England and Israel invade. Eisenhower is furious the countries have violated NATO alliance and failed to inform him of their plans. The USSR threatens with nuclear warheads until the three countries withdraw. NATO required the US to side with the invading countries, yet the US was not willing to go against the Soviet threat of nuclear attack.
In the end, Nassar was the only winner. He kept the Suez Canal, humiliated the former colonists, balanced the Cold War superpowers against each other, and he emerged as the leader of the Arab world.
Non-alignment has power. No longer were the super powers insured of getting their way. Their influence waned because a small country could defect to the other side or threaten to do so. The US and USSR struggle to keep everyone in their ideological playing field gave little powers the power. A new-to-the-superpowers expression emerged: Tails were beginning to wag dogs.
South and North Korea and Vietnam used the same manipulations. Ulbricht in East Germany threatened capitulation to the capitalists if he didn't receive more economic aid, more consumer goods and raw materials. Consequently, both powers supported puppet governments that compromised their ideals, and by supporting capitalistic democrats, who were also despots, Henry Kissinger later said, America had compromised her morals.