At any instant, the orbit of the Earth around the Sun lies in a plane. Clearly the Moon must lie in this plane for an eclipse to take place.
The orbit of the Moon around the Earth also lies in a plane. A different one, in fact. The intersection of these planes forms a line. The two points where this line meets the orbit of the Moon are known as nodes. In order to have a solar eclipse the Moon clearly must lie near a node.
As the moon orbits it travels from a node back to the same node in 27.21222 days (a draconic month).
A solar eclipse can only take place at a new moon. New moons take place every 29.53059 days (a synodic month).
Solar eclipses will take place each time the cycles of a half-draconic month (there are two nodes) and a synodic month coincide.
The ratio of the two periods is approximately 2.170391. So we can get an idea of when the cycles will coincide by constructing rational approximations to this ratio. We can use continued fractions to form them and one of these is 484/223. 242 synodic months is equal to 223 draconic months to within an hour and is equal to 6585 and a third days (approximately 18 years). This time interval is known as a saros.
So, if a solar eclipse has just taken place, then to a good approximation, we can expect another eclipse exactly one saros later. What's more, because a saros is 1/3 day, modulo a day, we know that the location of the subsequent eclipse will be 120 degrees longitude west. (Note that solar eclipses will happen more often than once a saros, but eclipses separated by a saros are interesting because they form a regular sequence, to a good approximation.)
But every rational approximation to the ratio given above will give some kind of approximate eclipse cycle, so why focus on the saros? The saros has another interesting property. The Moon's orbit is elliptical. This allows us to define another cycle: the time taken to move from major axis, to minor axis, to major axis again. This is the time period over which the Moon-Earth distance is periodic and is known as an anomalistic month. It turns out that that one saros is almost exactly 239 anomalistic months. When the Moon is closest to the Earth it looks bigger than the Sun and when it is furthest it looks smaller. This makes the difference between a total eclipse, where the Moon completely occludes the Sun, and an annular eclipse, where an annulus of the Sun is visible. Because a saros is close to an integer multiple of this period, solar eclipses separated by a saros are likely to be of the same type.
According to wikipedia this time period is named after a Babylonian word because the Babylonians were aware of this cycle. I'll believe that when someone tells me how the Babylonians knew that the eclipses were taking place one third of the way across the world.
Armed with this knowledge I must read how Stonehenge can be used to predict eclipses.