The Cold War led was a conflict between the United States of America (USA), along with its allies, and the Union of Soviet Socialist Republics (USSR) and its satellite states and allies. The deeper issues that set the USA and the USSR at odds 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. 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.
During the course of the Cold War, the two superpowers never directly went to war with one another. However, they did support a number of proxy wars where they were supporting opposing sides in other conflicts -- such as the Korean War, the Six Day War (in the Middle East), the Soviet-Afghan War, etc.
As the Cold War continued, the two superpowers (US and USSR) kept escalating their weapons capabilities and stockpiles. It got to the point that if the two sides did plunge into war, they would face mutually assured destruction. Eventually the US and USSR would pursue policies of detente, which included pledges to reduce their nuclear arsenals. The arms race and solving the arms race were constant issues affecting the Cold War.