The "Cold War" had that name because the two superpowers (the USA and USSR) did not fight a direct "hot" war against one another. They engaged in a protracted stand-off against each other, and had an arms race of nuclear weapons and military strength. They also supported "proxy wars" where they took opposing sides in conflicts happening in other parts of the world (such as the Six Day War in the Middle East).
Further detail:
Some of the deeper issues that caused rivalry and tension between the United States of America (USA) and the Union of Soviet Socialist Republics (USSR) were that the USA was committed to capitalism and democratic institutions of government, whereas the USSR was committed to communism and imposed authoritarian government. The Cold War was mostly a tension between these worldviews.
There were also immediate conflicts and pressure points as the Cold War began in the late 1940s (after World War II). One of those issues was that the USA had atomic weapons and the USSR did not. (The US would not share that technology with the Soviets, who had been their ally in World War II.) When the Soviets developed their own atomic weaponry, this led to a massive arms race between the superpowers -- which brought significantly more tension to the situation. Further tensions developed over which nations would be aligned with each of the superpowers -- either with the USSR and communism and with the USA and its democratic, capitalistic worldview. And (as mentioned above), the two superpowers aided differing sides in conflicts that took place in various parts of the globe during the Cold War years.
