While the superpowers have used whole governments as proxies, terrorist groups or other third parties are more often employed. It is hoped that these groups can strike an opponent without leading to full-scale war. Iran has used Hezbollah to this purpose to hit Israel and Pakistan has long used armed groups to advance its cause in Kashmir without going to war against India.
Proxy wars have also been fought alongside full-scale conflicts. During the Iran-Iraq War both nations armed factions in the Lebanese Civil War and pitted them against each other, for instance.
It is almost impossible to have a pure proxy war as the groups fighting for another power have their own interests, which are often divergent from those of their patron. For instance after the Soviet invasion of Afghanistan the groups the United States had been backing quickly turned and became the nucleus of the Taliban and al-Qaida.
A famous proxy war was the Spanish Civil War that saw an internal conflict become a battle between fascism and communism as Germany, Italy and the Soviet Union poured resources and advisors into the country. This proxy war served as a useful proving ground for the great powers to test equipment and tactics that would later be employed in the Second World War.
Proxywars were common in the Cold War, because the two nuclear-armed superpowers (the Union of Soviet Socialist Republics and the United States of America) did not wish to fight each other directly, since that would have run the risk of escalation to a nuclear war.
The war between the mujahadeen and the Red Army during the Soviet invasion of Afghanistan was a classic asymmetric war.
Of all the proxywars fought by the USA against the USSR during the Cold War this was the most cost effective and politically successful, as it was the USSR's most humiliating military defeat, and that defeat was a contributing factor to the implosion of the Soviet Union.
It is almost impossible to have a pure proxywar, as the groups fighting for a certain superpower usually have their own interests, which are often divergent from those of their patron.
This war served as a useful proving ground for the great powers to test equipment and tactics that would later be employed in the Second World War.
The first proxywar in the Cold War was the Greek Civil War, in which the Western Allied Greek government was nearly overrun by Communist rebels with limited direct aid from Soviet puppet states in Yugoslavia, Albania, and Bulgaria.