A customer service supervisor's job is to guarantee that customer experience at the organization they work for is like a day at the zoo, each and every time. And given the fact that people tend to be highly critical about customer care services, this is definitely easier said than done.
Typically, they rise to the position from a front-line representative position. Customer service supervisors oversee the work of customer care agent teams, coach the service staff, and make sure customer service teams are tightly bound and highly functional.
This is a managerial position, so you will need leadership skills. You need to be able to make people trust you and follow your vision, tactfully negotiate and resolve problems, and never let your frustration get the best of you. Leave that to the customers.