The proxy wars that I will reference are some of those that occurred after World War II and are considered a part of the Cold War. In each instance, these proxy wars reflected the conflict between the United States and the Soviet Union as it related to the spread of communism. While the United States and the Soviet Union didn’t directly fight each other, each country supported different sides in these conflicts.
In the Korean War from 1950-1953, the Soviet Union supported North Korea. North Korea, unprovoked, invaded South Korea in June 1950 with the goal of uniting Korea into a communist country. The United States, which supported South Korea, went to the United Nations to request action regarding this invasion. The United Nations created a multi-national force, led by the United States, to remove North Korea from South Korea. This war, which ended in a stalemate, kept South Korea free from communist rule.
The Vietnam War was another example of a proxy war. North Vietnam, supported by the Soviet Union, invaded South Vietnam after the government of South Vietnam, supported by the United States, refused to hold the mandated elections in 1956 to unite Vietnam. During the Vietnam War, we sent our troops to help South Vietnam fight against North Vietnam. After a cease-fire, which allowed the United States to leave South Vietnam, North Vietnam resumed the fighting and won the war. Vietnam became a communist country.
I assume that you are asking about proxy wars during the Cold War. These were wars (often civil wars) in which the US and the USSR would each support a separate side.
The significance of these wars was different for the superpowers than it was for the countries that were most directly involved. For the superpowers, a win or loss in one of these wars led to a relatively minor change in power and prestige. For example, the US loss in Vietnam had a fairly minor impact on the United States. By contrast, the wars were typically devastating for the countries in which they happened. They caused destruction as well as lingering enmity between the combatant groups within the countries.