It appears easy to understand why there should be a bulge of water, producing a high tide, on the side of the earth facing of the moon. But why is there a bulge on the opposite side as well?
It is obviously not gravity that is doing it but rather, it is the difference in gravitational force across the earth that causes the bulge. This difference in gradational force comes from the moon's pull at various points on the earth.
Because the pull of gravity becomes stronger as the distance decreases, the moon pulls a little harder at point "C" (closest point to the moon) than it does at point "O" (in the center of the earth), and the pull is weaker still at point "F" (farthest point from the moon). If it were not for the earth's gravity, the planet would be pulled apart (above image).
Yet also because of the earth's gravity which pulls us toward the center of the planet we can, mathematically subtract the moon's pull at the center of the earth from the moon's pull at both point "C" and "F". When this vector-based subtraction occurs we are left with two smaller forces; one toward the moon and one on the opposite side point away from the moon (image below) producing two bulges.
As the earth makes one rotation in 24 hours, we pass under these areas where the tidal force pulls water away from the earth's surface and experience high tides. Also, since the difference in gravitation force is constant across the earth, the bulge on both side of the earth is essentially the same. Which explains why consecutive high tides are nearly the same height each time regardless whether the moon is overhead or on the opposite side of the earth.