As both the United States and the Soviet Union possessed extensive nuclear capabilities, their respective geopolitical goals needed to be achieved without direct confrontation. Thus it became necessary to conduct the Cold War to a large extent through proxy states. As well as potentially avoiding the dangers of nuclear armageddon, this strategy enabled the United States and the USSR to advance their ideological agendas. The United States could, through developing relations with their proxy states, allow the opening up of new markets to American industry. Such partners could take various forms, whether they were military dictatorships in Latin America or European liberal democracies.

The Soviet Union, for its part, increasingly posed as the defenders of countries in the developing world in their anti-colonialist struggles. Although there was some ideological underpinning to the Soviets' strategy—the spread of international Communism among the world's oppressed masses—there was also a fair amount of opportunism involved. Successive American governments had propped up a variety of repressive dictatorships in the developing world, allowing the Soviets to exploit any guerilla or resistance movements.

On paper, the strategy of using proxies seemed sound enough. However, in practice, the results were often far from impressive. Much depended on the degree of control that the United States and the USSR could exert over their client states. In Vietnam, for instance, Diem's increasingly high-handed approach to domestic problems, in particular his persecution of Buddhists, led to a growing strain with the American government. Diem's exacerbation of an already tense political situation undoubtedly helped to give rise to the conditions for America's subsequent military involvement in Vietnam.

The Soviets experienced much the same problems. Khrushchev assiduously cultivated relations with the Castro regime in Cuba. At first all seemed well, but the deployment of Soviet missiles within striking distance of the United States took the world to the brink of nuclear destruction during the Cuban missile crisis. Throughout the crisis Castro had constantly exhorted Khrushchev not to back down, so when he subsequently did, Soviet-Cuban relations soured considerably. Although relations did eventually improve, the incident seriously impaired the ability of the Soviet Union to achieve its goals in the American sphere of influence.

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.