The term “proxy war” is a term from the Cold War. It refers to wars that were essentially wars between the United States and the Soviet Union but which were actually fought by proxies on at least one side of the war. Proxies are people who stand in for others, doing something in the place of the other person.
During the Cold War, the US and the Soviet Union were enemies. However, they did not feel safe fighting one another in wars as enemies might be expected to do. The reason for this is that they feared that a war between them would escalate into a nuclear war, destroying both countries. To avoid this, one or both sides would have other countries or other groups fight for them. That meant that they would not directly confront one another.
One example of this type of war is the Vietnam War. In the early stages of the war, the US and the USSR both fought through proxies. The US had the French as their proxy while the Vietminh were the proxy for the USSR. Later on, the US entered the war on their own, but the Soviets did not do the same. Instead, they continued to use the Vietcong and North Vietnam as their proxies. In Afghanistan, the shoe was on the other foot as the USSR itself fought during and after the invasion of 1979 while the US used “mujahedeen” fighters as its proxies.