What did the Soviet Union want during the Cuban missile crisis?

The soviets wanted a military base in Cuba.

Explanation:

During the Cuban missile crisis the essential demand of the soviets was that they wanted to make a military base in Cuba that can house nuclear weapons.

This was in response to the US military bases that were in central Asia as the soviets felt the need for something as dangerous for the US to be there in their vicinity too.

So the conflict arose out of a defensive strategy that required both parties to project strength.