The Cold War ended with the collapse of the Soviet Union which was brought on by the reforms of Mikhail Gorbachev. The end can be said to have come in 1989 when the Berlin Wall fell. It was complete by 1991 when the Soviet Union broke up.
The underlying cause of the end of the Cold War was the communists' inability to keep their people happy. The communists' centralized economy could not make enough consumer goods of the sorts that people wanted. This led to a great deal of unhappiness among the people of the communist bloc.
The end of the Cold War was pushed along by the actions of Gorbachev and President Ronald Reagan. Gorbachev attempted to open the communist economy and to give people more political rights. When given an opening in this way, the people pushed harder than expected and communism fell. Reagan helped bring this about by pushing for an arms race with the Soviets. By expanding US weaponry, Reagan forced the Soviets to respond, further weakening the Soviet economy's ability to make consumer goods and keep the people happy.