The distance between the Earth and the Moon is not constant, but varies over the course of the Moon's orbit around the Earth, which is not a perfect circle but an ellipse. The point at which the Moon is furthest from Earth is called the apogee, and the nearest point is called the perigee. These points do not coincide with the phases of the Moon (i.e. when there is a full moon, or crescent moon, etc), and they vary from year to year.
The average distance, from the center of the Earth to the center of the Moon, is 384,403km (238,329 miles), with the distance at perigee being 363,104km (225,124 miles) and at apogee 405,696km (251,531 miles). As a basic rule of thumb, to say that the moon is a quarter of a million miles away is a good enough estimate.
The calculation of the distance was made considerably easier, and to a greater degree of accuracy, when the Apollo astronauts left reflective plates on the Moon's surface in the early 1970s. Three reflectors were left at various sites. Daily checks are now made by sending laser signals from Earth observatories and measuring the time taken for the signals to hit the reflectors and be received back on Earth. The returned pulses are extremely weak, and there are only three observatories that are equipped to detect these signals and convert them into accurate calculations of the Earth-Moon distance. There could therefore be inaccuracies caused by such things as slight fluctuations in the Earth's shape that are not being picked up by the current arrangements.
Scientists are hoping that a new manned mission to the Moon will be able to replace the reflectors, which have possibly suffered damage from tiny meteorites in the 30 years they have been there, meaning that the bounced signals are much weaker than they once were, with transponders that will not just reflect the laser beams but send pulses of their own. These stronger pulses could be picked up and interpreted by many more Earth observatories, and more accurate results be obtained.
However, even given these problems with the current methods, we know that the Earth-Moon distance is increasing by about 3.8cm (1.5 inches) a year. This is caused by gravity interactions between the Earth and the Moon, such that the Earth is pushing the Moon away from it at every revolution. By extrapolating backwards in time, we can therefore work out that the Moon must have much closer in the distant past, and indeed that the two bodies are in reality a double planet.
This means that, a billion years ago, the Moon would have been 10% closer to Earth than it is now. However, other evidence, based on geology, suggests that the Moon's recession has speeded up, and that it was probably only receding at 2.0cm a year at that time.
Looking the other way, we can calculate that at some future time the Moon will have drifted so far out, and slowed down to such an extent, that it will not appear to orbit the Earth at all and will simply hang in the sky over one point and not be visible at all from much of the planet. However, the Sun will have gone out by then and there will be no-one around to witness this!
One remarkable thing about the current Earth-Moon distance is that the orb of the Moon in the sky appears to be almost exactly the same size as that of the Sun, which is of course many times larger than the Moon but at a considerably greater distance away. Therefore, when we see a solar eclipse, caused by the Moon coming between the Earth and the Sun, not only is the light of the Sun blocked off for the short time of the full eclipse, but we are able to witness the full glory of the solar corona (using all the correct viewing precautions, of course). In the very remote past, when there would have been no humans to witness it, a solar eclipse would have meant complete darkness for much longer, and no view of the corona, as the Moon's disc would have been larger than that of the Sun.