Prior to the 20th century, the best way to determine the distance to the Moon was to use parallax. The Moon's coordinates in the sky are slightly different from different places on Earth. The Moon forms a skinny triangle with the two locations on Earth. The small angle is the angular distance between the two measured lunar positions. Its opposite side is the physical distance between the two places on Earth. Using trigonometry, you can thus find the distance to the Moon.

With the invention of RADAR, more precise determinations of the lunar distance became possible. Also, thanks to the Apollo astronauts, there is a large reflector on the Moon's surface, off of which laser light can be bounced from Earth. By measuring the time the laser light takes to make the round trip to the moon and back, its instantaneous distance can be measured to within one inch (!).

Average distance to the moon: 384,400 km.