The "nuclear button" is a figurative term referring to the power to use nuclear weapons. "Pushing the nuclear button" refers to actually using them. The actual procedure for using such weapons is more complex than simply pushing a button. The "nuclear button" may be transferred to another official due to political changes or the incapacitation of a person currently in control of it.
Various nuclear countries have nuclear briefcases that accompany the leader (such as the president of the United States) allowing him or her to launch nuclear missiles at any time.