The first measurement of the distance from Earth to the Moon and the Sun made a Greek astronomer Aristarchus of Samos (310-230). He started from the observation that the diameter of Earth's shadow falls on the moon during a lunar eclipse is about two times larger than the diameter of the moon. (more accurate value is about 2.6 times.)
Modern measurements of the distance of the Moon are performed to measure the time it takes radar or laser light pulse to reach the Moon, reflects and return to Earth. Knowing that time and the speed of light, it is easy to calculate time and divide by two.
The greatest distance of the Moon from Earth (apogee) is 405 000 km, the mean distance is 384 000 kilometers, and a minimum distance (perigee) is 363 000 kilometers. The difference in distance occurs because the moon (and most other celestial bodies) moving around the Earth in an elliptical path. It is interesting that the ancient Greeks had already managed to measure its distance. Contemporary modern measurements are made using radar, and since they were on the the moon, they left laser reflectors with which the laser can measure the distance of the Moon accuracy up to several centimeters. Specifically, the laser beam "fire" reflector on the Moon and then the same beam flows back to Earth. Using a precision measuring instruments, monitors the time it takes to get the beam back to Earth, and thus calculate distance from Earth to Moon.
Last month something happened that numerous media called "Super Moon". Although with the emergence of speculation as the cause of many events on Earth, the reality is much different. Our natural satellite Moon orbits the Earth in an ellipse. When it is closest to Earth separates us distance from over 350,000 km and when it is furthest more than 400,000 km.