The ancient Greeks had a number of methods for determining the distance to the moon. Aristarchus was the first person to determine how far away the moon was. Based on observations of a lunar eclipse and on observations of a quarter-phase moon, he was able to calculate the relative sizes and distances of the earth, moon and sun. Hipparchus used parallax data from a solar eclipse to also calculate how far away the moon was. It is important to remember that the Ancient Greek calculations were all geometrically valid and correct - the only limit to their accuracy was in their measurements - of positions in the sky, locations on the earth and of time. They were only limited by the technology of their time.
Now, high precision measurements are made through the use of mirrors left on the surface of the moon from the Apollo missions (see http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_Experiment Special reflectros were left on the surface of the moon in a few different locations. They are designed to return a beam of light back on itself. Using high-ppowered lasers and telescopes, a pulse of light is fired towards the moon targeting one of the reflectors, and the exact time delay for the return signal is measured. Since the pseed of light is known, the classic "distance = rate times time" can be used.
The picture below is a scale image of the earth and moon taken from http://en.wikipedia.org/wiki/File:Earth-Moon.png