There were other countries that avoided becoming involved in the Cold War the same way they avoided involvement in World War II - they were simply permanently neutral. Switzerland prides itself on both its neutrality and peaceful stability. They have not been to war in over six centuries (which is one reason why they are an attractive place to invest money) and are blessed with a geography that allows them not to take sides.
Sweden is another country that has remained stubbornly neutral for a century or more. When the Cold War began, they simply stayed the course of neutrality they had previously set, and it worked well for them.
The United States and the Soviet Union were the two "super powers" of the world during this time period. These two countries also tried to antagonize one another by threatening to use weapons. In particular, nuclear weapons. Most countries wanted no part of this so they stayed neutral. Both countries had allies but most countries tried to stay out of the way.
Many countries also refused to become part of NATO (North Atlantic Treaty Organization) or the Warsaw Pact. Because of this they were proclaiming no allegiance to anyone, therefore promoting neutrality.
The most common thing that countries did to try to avoid becoming involved in the Cold War was to declare themselves to be nonaligned. Bascially, they would declare that they were not going to be part of it.
A few countries managed to pull this off. Egypt is one of the major countries that managed to do this. The other two countries associated with the nonaligned movement were Yugoslavia and India.
These countries generally tried to pursue policies that they thought were important to the smaller countries of the world -- policies that were not involved in the superpower conflict.
To avoid getting embroiled in the rivalry between USA and USSR that developed after the Second World War, which is generally called the Cold War, many countries choose to explicitly maintain and announce their neutrality toward these issues. They declared themselves as "non-aligned" and also formed a group of non-aligned nation to form common approaches for dealing with the rival groups of the cold war.
The person that took leading role in formation and and conduct of the group of non aligned nation included Prime Minister of India Nehru, President Nasser of Egypt, and President Tito of Yugoslavia. The activities of the group of Non-Aligned Nations started with a conference held in Indonesia in 1955, which was attended by 29 Asian and African countries. By the time the 11th summit of Non-aligned nation was held in Colombia in 1995, its membership had increased to 113 countries.