Eclipses are, of course, caused by the orbit of the Moon around the Earth. Since the Moon goes around the Earth once a month, this seems fairly simple! But, as described in Cycles of Eclipses, things are quite a bit more complicated than that.
The problem is that the idea of a month isn't so simple: in fact, in astronomy, there are no fewer than five kinds of month! Most of these have some relevance to eclipse prediction, and so this page will try to explain them in more detail.
The Moon's orbital period is the time it takes for the Moon to complete one orbit around the Earth; ie. for it to go from "due West" of the Earth, say, in the diagram below, once around the Earth, and back to "due West" again. This period is known as a Sidereal Month, and is 27.321661 days.
Seems pretty simple — but it isn't, quite! For example, you might think that in a sidereal month, the Moon would go from a New Moon to the next New Moon. As you can see in the diagram, though, it doesn't work; in the time it takes the Moon to orbit the Earth, the Earth itself has moved, so that the Moon isn't between the Earth and the Sun when the sidereal month is up — and, therefore, isn't a New Moon. So while the sidereal month seems like the simplest view of what a "month" should mean, astronomically, we'll see that the other kinds of month are actually more interesting from an Earthly point of view.
Rotational Period: The rotational period of the Moon is the time it takes for the Moon to complete one rotation on its axis. Interestingly, since the Moon's rotation is synchronised, by tidal forces, with its orbit around the Earth, the rotational period is exactly the same as the orbital period: 27.321661 (Earth) days. Because of this, the Moon always keeps the same face turned towards us (give or take a few degrees of variation, caused by the Moon's orbital inclination).
The far side is therefore never visible from Earth, and is consequently known as the "dark side" — "dark" in the sense of unknown, obscure, hidden, etc.; it wasn't until the Soviet spacecraft Luna 3 photographed it in 1959 that we got our first view of the Dark Side of the Moon. By now, of course, the whole Moon has been thoroughly mapped and photographed by orbiting probes and astronauts.
Bear in mind that the Moon does, in fact, rotate, and because of this every spot on the Moon sees day and night in the same way as on Earth — albeit that the lunar days and nights are each 14.8 Earth days long. So there is no permanent "dark side of the Moon" in terms of illumination.
Note that the rotational period is not the same as a lunar day!
The Moon's orbit around the Earth is not, in fact, circular; it's elliptical. Only slightly, but enough to mean that the Moon isn't always the same distance from the Earth.
Two points in the Moon's orbit are of special interest:
The points of apogee and perigee are not fixed relative to the Earth. This is due to regression of the Moon's orbit; the "long" dimension of the elliptical orbit, and hence the points of apogee and perigee, circle around the Earth about once every 9 years. Thus, the time it takes the Moon to travel from apogee to perigee and back again is slightly longer than its orbital period (the Sidereal Month). This period of time is called an Anomalistic Month, and is 27.554549 days.
The Anomalistic Month is interesting for solar eclipses, because the "size" of solar eclipse that we see — and the type of eclipse, whether it is partial, total, annular, or hybrid — depend on the distance from the Earth to the Moon during the eclipse; and this distance depends on the point during the Anomalistic Month when the eclipse occurs. For lunar eclipses, the duration of the eclipse, and the appearance of the eclipse, are also affected by this.
But for actually predicting the occurrence of an eclipse, we need two more kinds of month!
As we described in The Earth and Moon, a very clear cycle is visible when you look at the Moon from the Earth: the cycle from New Moon, through crescent, half and gibbous Moons, to the Full Moon, and back again. It may seem that this cycle should be the same as the Sidereal Month, but once again, things aren't so simple!
The problem is that as the Moon is orbiting the Earth, the Earth is going around the Sun; and while the Moon is busy completing its orbit of the Earth, the Earth moves a twelfth of the way around the Sun. As you can see in the diagram above, this means that starting from a New Moon, the Moon has to go around a full orbit and a bit more to get back in between the Earth and the Sun again.
The time it takes for the Moon to go from one New Moon to the next is called a Synodic Month, and is 29.530589 days on average. Because the orbits of the Earth and Moon aren't circular, and hence the two bodies don't move at a constant speed, the actual time between lunations may range from about 29.27 to about 29.83 days.
Lunar Day: The length of a day on the Moon is the time it takes for the Sun, as seen from an observer standing on the Moon, to go from overhead to overhead. Again, this is not the rotational period of the Moon, because the Moon has moved round the Sun during that period; so, a Lunar day is the same as the time it takes for the Moon to go from full to full: ie. one Synodic Month, or 29.530589 days.
One thing that may occur to you: if the Moon's day isn't the same length as its rotational period, what about the Earth? The answer is that the same thing applies on Earth: because the Earth is orbiting around the Sun at the same time as it's rotating, a sidereal day on Earth — which is one complete rotation of the Earth on its axis — is 23 hours, 56 minutes, 4.06 seconds, which is less than a solar day of 24 hours.
The diagrams above show the Earth and Moon as seen from above; in this view, things look pretty simple and two-dimensional. Unfortunately, reality is just a little more complicated than this. To understand the next type of lunar month, we'll need to take a slightly different view.
As described in Cycles of Eclipses, the Moon's orbit is tilted to the plane of the ecliptic; the two points at which the orbit intersects the ecliptic are known as nodes. In order to get directly between the Earth and the Sun, the Moon has to be in the plane of the ecliptic; and therefore has to be at one of the two nodes. So the time the Moon takes to travel from a node, around its orbit, and back again, is of great interest.
Once again, this period is not quite the same as a single orbit, due to precession of the Moon's orbit; the Moon's orbital plane, and hence the nodes, rotate backwards around the Earth about once every 18.6 years. The time taken for the Moon to return to a node is therefore shorter than its orbital period; this time is known as a Draconic Month, and is 27.212220 days. This is sometimes called the Nodal Month.
Incidentally, the term "Draconic" refers to a mythological dragon, that supposedly lives in the nodes and regularly eats the Sun or Moon at an eclipse.
So where does all this leave us? Well, as we've seen, an eclipse happens when the Moon is at one of the two nodes; and the period which determines this is the Draconic Month. However, the Moon also has to be aligned with the Earth and the Sun, ie. at a New Moon (producing a solar eclipse) or Full Moon (for a lunar eclipse); and the period governing this is the Synodic Month. When these two cycles coincide, an eclipse occurs!
Now, it happens that 223 Synodic months is 6585.321 days; and 242 Draconic months is 6585.357 days, which is almost the same, and indeed close enough to produce an eclipse. This period, of 18 years, 10 and a third days, is known as the Saros; this is the period at which similar eclipses (ie. eclipses belonging to the same Saros cycle) re-occur.
The slight difference between the two totals is why successive eclipses in a Saros series do, in fact, differ; the eclipses in a series migrate gradually from North to South or vice versa.
The odd third of a day (roughly) is the reason why successive Saros eclipses occur over completely different parts of the Earth: the Earth has rotated a whole number of days plus a third between the two eclipses. On the other hand, this means that 54 years and 32 or 33 days (depending on leap years) after an eclipse, another eclipse in the same Saros series will occur, and the Earth will have rotated back roughly to where it started. This leads to another eclipse cycle — the Triple Saros, which is three times the Saros period. This is the interval over which similar eclipses will occur over the same side of the Earth, but shifted significantly North or South.
Finally, there are many different combinations of circumstances which can cause an eclipse: Moon at its rising node, Moon at the falling node, etc.; and this is why there are 42 Saros cycles running at any one time, and hence why we get several eclipses (solar and lunar) per year.
Just for completeness, one more month you might come across is the Tropical Month, which is the period from one lunar equinox to the next, and is 27.321582 days.