The Klein-Gordon Equation or the Klein-Fock-Gordon Equation is an equation in quantum field theory which initially was discovered by Schrodinger but discarded by him soon after.
The Klein-Gordon Equation is an equation in field theory initially discovered by Schrödinger (before he introduced the equation that carries his name) but discarded by him soon after. The equation reads $$ (\partial^2-m^2)\phi=0 $$ where $m$ is a real parameter (typically identified with a mass), and $\partial^2$ is the Minkowski Laplacian.
This is in natural units of course. In SI units, or any other system of units, the mass needs to be scaled by the ratio of the speed of light to the reduced Planck constant.
This equation, however, does not predict a conserved probability current with positive-definite charge (because of it being second-order in time). This is why Schrödinger discarded the equation. However, it has been resurrected in Quantum Field Theory to describe scalar fields, where the conserved charge is to be thought of as an electric charge rather than a probability amplitude.
The problem with the Klein-Gordon Equation is partially solved by the Dirac Equation, which is first-order in both space and time. Nevertheless, the status of the former as a point-particle wave equation is challenged by other difficulties so, once again, the equation is only meaningful in the context of quantum field theory.