What is it called when the moon is closest to earth?

The Word "perigee" means the point in an orbit when the satellite is as close as it ever gets to the Earth. The prefix "peri" means "near to", and "gee" is used to refer to the Earth's gravitational mass.

The Word "perihelion", from "peri" for near and the Greek name of the Sun "HeliOS", means "as close to the Sun as something ever gets".