To pivot is to turn or rotate, like a hinge. Or a basketball player pivoting back and forth on one foot to protect the ball.
When you're not talking about a type of swiveling movement, you can use pivot to mean the one central thing that something depends upon. The central pivot of your student government campaign, for example, might be longer recess time. Or your plans to go to the beach this weekend could pivot upon whether it rains, as the weather channel has predicted, or gets surprisingly sunny.