The nuclear arms race between the superpowers ended in the 1980s because the Soviet Union lost the ability to compete with the US and finally collapsed. This came about as the Soviets tried to open their economic and political system a bit in the Gorbachev era.
In the early 1980s, after detente had fizzled, a new arms race broke out. President Reagan pushed this arms race because he believed that this was the best way to deal with the Soviets. In the end, it appears that this arms race broke the Soviets. They were under tremendous pressure to keep up with the US and their economy could not stand up to the strain. Because of this, they lost the ability to both keep up in the arms and keep their people at least somewhat content.
When the USSR lost this ability, it was unable to continue in the arms race and that race ended along with the reign of the Soviet Union.