Reply of 9/24/2007 (NOT to be considered a "textbook" discussion)
     Although there are a number of popular myths connecting earthquakes and the Moon, careful (i.e., statistically valid) studies show that there is no correlation with the time of day, the position of the Moon in the sky, the phase of the Moon, or the distance of the Moon from the Earth. There is a relationship between the distance of the Moon from the Earth, and "moonquakes", or seismic activity on the Moon. Most (if not all) lunar quakes are caused by tidal stretching of the Moon, which averages about 30 feet (about 20 times greater than the average Earthly tide of a foot and a half), and can increase or decrease by about 20% of that 30 feet (or about 6 feet) due to changes in the lunar distance.
     The math works out like this:
Starting facts:
     Earthly tides average about 1 1/2 feet (based partly on the theory of tides, and partly on observation)
     The Earth and Moon exert the same overall gravitational force on each other (Newton's 3rd Law of Motion, the Law of Action and Reaction)
     The Earth is about 80 times more massive, and four times larger diameter, than the Moon (observation)
Reasoning and Results:
     Since the Earth exerts the same force on a body 80 times less massive, it would produce 80 times larger tides on the Moon than the Moon does on the Earth, all other things being equal. But tides are caused by tidal forces, which are the difference between the gravitational force at a given place, and the force at the center of the body (which equals the average force on any part of the body), and the Moon being four times smaller cuts down the effect of our larger force per unit of mass by a factor of four. Hence the final result, that lunar tides average 20 times larger than ours, or about 30 feet.
     The lunar tides do not move around the Moon, because the Earth doesn't "move" around the Moon. Since the Moon always keeps one face to the Earth, the Moon experiences "frozen" tides, in which the tidal stretching (which always points toward and away from the Earth) is greatest at the centers of the near and far sides of the Moon.
     If the Moon kept a constant distance from the Earth, the tidal stretching would be fixed, as well as "frozen"; but it gets up to 6% closer or further to/from us than on the average, as a result of its orbital eccentricity. This increases/decreases the overall force by 12% (since gravity is an inverse square law), and the tidal force which stretches the Moon by 18% (because the Moon looks bigger when its gravity is stronger, due to being closer). Hence the approximately 20% change of about 6 feet in the size of the lunar tides.
     As a result of the 6 feet / 1000 mile stretching, the Moon suffers weak earthquakes, averaging about half the depth of the lunar radius, which are strongly concentrated around the time of its closest approach to the Earth (perigee), every 27.3 days.
     Now, one could argue that if the Earth can do this to the Moon, the Moon ought to be able to do it to the Earth, as well, but as it turns out, it cannot, for two reasons:
     (1) As stated, the force on each body is the same, but the tidal force/stretching on the Moon is 20 times larger than that on the Earth, because of the difference in their size and mass. If we were to ask what kind of seismic activity we see on the Moon when the stretching is only 1/20th of the maximum, and is equal to the typical tidal stretching of the Earth, the answer would be: none
     (2) Also, tidal stretching/force does not occur at the surface of a body, but throughout its interior. In that part of the Moon which experiences "moonquakes", the overall depth is about half of its radius of 1000 miles. Hence, close to 50% of the tidal stretching occurs there, or about 3 feet of stretch in 500 miles. That is what is causing the moonquakes. In the case of the Earth, the tidal stretching is 20 times smaller, but the area where earthquakes normally occur -- the lithosphere -- is only about 1% of the Earth's radius (about 40 to 50 miles, out of 4000 miles), so the stretching in that region is only 1% of 1 1/2 feet, or 1/2% of the lunar stretching, over a distance only ten times smaller. Correcting for the smaller region, the average stretch within the portion of the Earth subject to normal faulting is only about 2% of the stretching that is going on, on the Moon, when moonquakes occur.
     In other words, although the powerful tidal forces exerted on the Moon by the Earth can easily cause weak quakes in the lunar interior, the much smaller stretching of the Earth by the Moon in its crust/lithosphere is far smaller, and cannot be expected to exert any influence on Earth quakes.
     That doesn't mean, of course, that it isn't possible for some subtle mechanism to amplify the lunar tidal effect, and cause earthquakes to be somewhat correlated with some aspect of the lunar position(s). So many studies have been done, looking for such correlations. And as is true of any large group of studies, some of them should yield apparently meaningful results, just by chance (e.g., if there is a 1 in 25 chance that an apparently significant result would result by chance, then 1 in 25 studies will have an apparently significant result, even if there isn't any real connection). Needless to say, it is the "chance" studies that are trumpeted in the press, hence the general misconception that the Moon has some effect on earthquakes. But if such an effect were at all real, geologists interested in predicting earthquakes would gladly use it. So there have been lots of studies, all of which (save for the occasional random chance "flag") show no statistical correlation between (1) the position of the Moon, relative to perigee (there should be more quakes when it's closer, but there aren't), (2) the phase of the Moon (the solar and lunar tides add up at full and new moon, so there should be more quakes then, but there aren't), (3) the position of the Moon in the sky (rising or setting, tidal forces might compress the crust, while overhead or underfoot, they might stretch it; the effects are unknown, but if significant, should yield non-random results; and they don't), (4) and so on and so forth.