The USSR and USA were at odds with each other throughout the 20th century, especially during the Cold War. Each country was a world superpower with very differing views. The USSR was communist and a "threat" to all things that were good and capitalist. The USA was a "threat" to communist because of its evil ideology. Since both of these nations had a nuclear arsenal at their disposal, war would have ended life on Earth as we know it. However, neither side was willing to give up their weapons- power and security were very important to them.
This made it almost impossible to come to any sort of agreement. The USSR was supplying third world countries with weapons and support- with the intention that those countries would become communist. Cuba, being so close to the US (and Washington, D.C.) was a prime spot for building a Soviet nuclear base. The USSR was on its way to deploying weapons to Cuba, and the US came very near to nuclear annihilation. In the 13 day standoff, Kennedy agreed to allow Cuba sovereignty if the Soviets would withdraw their weapons from Cuba- which they did.
The United States and the Soviet Union came to the brink of nuclear war in 1962 over Soviet missile bases in Cuba. This was referred to as the "Cuban Missile Crisis". The US was mostly concerned about the very close proximity of the missiles to its shores.
