The periodic table as we know it today is managed by the International Union of Pure and Applied Chemistry, or IUPAC (eye-you-pack).
While much of what is in the periodic table is stable and unlikely to change, the IUPAC organization is responsible for deciding what needs to be changed. They have created criteria for what constitutes the discovery of a new element.
In addition, any new element must be assigned a temporary name and symbol, and if validated, given an official name. Such was the case when IUPAC recently reviewed elements 113, 115, 117 and 118, and decided to give them official names and symbols (goodbye, ununseptium and hello, tennessine!).