As described previously, an eclipse occurs when the Moon comes between the Earth and the Sun (a solar eclipse), or the Earth comes between the Moon and the Sun (a lunar eclipse). Since the Moon orbits around the Earth, as we described in The Earth and Moon, it's easy to see how this can occur. However, if it was that simple, there would be a total solar eclipse once every month, at the New Moon, and a total lunar eclipse every Full Moon; and this isn't the case. So, in fact, it must be more complicated than that; and sure enough, it is!
The good news is that it's not too complicated; and in any case, you don't have to understand any of this to enjoy an eclipse! But if you're interested in the mechanics behind an eclipse, read on.
First, a little terminology. As the Earth circles around the Sun (actually, the Earth's orbit is slightly elliptical, but never mind for now), it stays within a flat plane, known as the ecliptic. When imagining the workings of the Solar System, it is conventional to think of the plane of the ecliptic as being horizontal, with the North Pole of the Earth pointing "up".
Think of it like this: put an orange in the middle of a table to represent the Sun, and a grape towards the side of the table representing the Earth. As the Earth moves in its orbit around the Sun, it stays on the table top; so the table represents the ecliptic.
This is exactly the same arrangement we described in The Earth and Moon; all we've done so far is added the name of the ecliptic plane. Now, though, we have to add one little wrinkle we haven't seen yet — the Moon's tilted orbit.
Now, you can represent the Moon's orbit by putting a pea near the Earth, circling around it (although the Moon's orbit is actually even less of a circle than the Earth's). Fair enough, it seems; the Moon orbits the Earth in the plane of the ecliptic, just like the Earth orbits the Sun.
Unfortunately, though (for simplicity's sake), this isn't actually true; the Moon's orbit is, in fact, tilted slightly off the ecliptic. The centre of the Moon's orbit is the Earth, of course, which is on the ecliptic; so as the Moon orbits, it alternately dips below, and then rises above, the ecliptic.
This diagram (which is, again, on a wildly exaggerated scale) tries to illustrate this:
Here the ecliptic plane is represented by the translucent green and blue checkerboard, with the Sun in the centre, and the Earth moving in its orbit within the ecliptic; the Earth's orbit is shown in blue. The plane of the Moon's orbit is shown as the tilted red and yellow checkerboard (in reality, it's only tilted by about 5 degrees), with the Earth being the blue and white ball in its centre. The Moon is shown in white, and its orbit is shown as a white line.
You'll notice two red blobs drawn on the Moon's orbit. These represent the point where the Moon's orbit crosses the ecliptic plane, and are referred to as nodes. Now, an eclipse can only occur when the Moon is in line with the Earth and Sun; but a line from the Earth to the Sun — drawn here in purple — lies along the ecliptic. So, an eclipse can only occur when the Moon is in (or near) the ecliptic; and that means that it has to be at, or near, one of the two nodes, and that node has to be positioned in line with the Earth and the Sun.
As the whole Earth-Moon system orbits around the Sun every year, the two nodes will find themselves aligned with the Earth and Sun (one in between, and one "behind" the Earth) twice a year; this means that there are two times each year when we can get an eclipse. Because the node doesn't have to be exactly lined up to cause an eclipse, there is actually a period of between 31 and 37 days during which an eclipse can occur. These times — when one of the Moon's nodes is approximately in line between the Earth and Sun, so there is the possibility of an eclipse — are called eclipse seasons.
As the Moon goes round the Earth, if it passes through either node during an eclipse season, an eclipse will occur; a solar eclipse if the new Moon passes the node between the Earth and Sun, or a lunar eclipse if the Full Moon passes the other node. Furthermore, if the Moon is (more or less) exactly at that node at the middle of the eclipse season, the eclipse will be total (or maybe annular, for a solar eclipse) as seen from some part of the Earth.
This diagram shows the Moon passing through a node, while that node is between the Earth and the Sun; this leads to a total solar eclipse. I've cut out the middle of the Moon's orbital plane so you can see the Moon's shadow falling on the Earth:
As a matter of fact, the Moon's orbit itself is gradually rotating on its axis, with the effect that the nodes gradually rotate around the Earth. For this reason, an eclipse season happens a little more often than every six months; in fact, every 173 days.
OK, so we get eclipses every 6 months (or slightly less), right? Right! But what we don't get is identical, or even similar, eclipses every 6 months. For example, as you can see from my list of eclipses from 2001–2020, the hybrid eclipse of 3 November 2013, up in the northern hemisphere, is followed 6 months later by a tiny annular eclipse on 29 April 2014, visible in the low southern hemisphere. In general, if you look at the eclipse maps there, you'll see that successive eclipses are scattered all over the globe. So how come?
Well, the answer is that there are even more cycles at work. For example, the time the Moon takes to go from New to New (a Synodic Month) is different (due to the movement of the nodes) from the time it takes to travel from one node, around its orbit, and back to the same node again (a Draconic Month).
The end result of all of this is that these various cycles mesh together to produce the same set of circumstances — and hence a similar eclipse — every 18 years and 10 or 11 and a third days. (Whether it's 10 or 11 days depends on how many of the 18 years are leap years). Amazingly enough, this period was actually discovered 2,500 years ago, by Babylonian astronomers; it's called the Saros, meaning "repetition".
A Saros series is, then, a series in which similar eclipses happen every 18 years 10/11 and a third days. But of course, eclipses happen more often than that; the explanation is that there are many combinations of circumstances that can produce an eclipse. So, there are, at any one time, 42 Saros cycles running at once; this results in a bit more than 2 eclipses per year.
Each Saros series has a number to identify it — for example, the 11 August 1999 eclipse belongs to Saros series 145. Eclipses within a Saros series are similar to each other, but different to eclipses of other series — they happen in different parts of the Earth, or are partial as opposed to total, etc.
The Saros cycle itself isn't perfect; the various lunar cycles don't quite mesh up perfectly. For this reason, successive eclipses in a Saros series are shifted slightly either north or south (depending on the particular Saros) from each other. This means that a Saros series is actually of limited duration — about 70 to 85 eclipses over 1,200 to 1,500 years. Each series starts with a small partial eclipse in either the north or south polar regions; as the shadows of the successive eclipses move farther into the Earth, the first total eclipse will be seen near the polar regions. The eclipses of the series then march down or up the Earth, until the last total eclipse, and then a series of diminishing partial eclipses, occurs at the opposite pole to where the series started; and then it ends.
The fact that the length of a Saros has an odd third of a day means that successive eclipses don't occur in the same part of the world; since the Earth has rotated by an additional third between eclipses. This gives rise to an even longer cycle, the Triple Saros, in which eclipses occur in the same part of the world every 54 years and 32 or 33 days (depending on leap years).
So can we see this at work? Yes, if you look carefully at the solar eclipse maps, you'll see this happening; each eclipse is followed by a similar one, 18 years and 10 or 11 days later. Bear in mind, though, that while all this is going on in space, the Earth is also rotating; because of this, each successive eclipse within a Saros series occurs about 120° west in longitude from the previous one in the series. Hence you can see the Triple Saros, where similar eclipses occur every 54 years.
So, look for example at the map of eclipses from 2001–2020. The first total eclipse on this map, on 21 June 2001, is followed, 18 years later, by the one on 2 July 2019, which is about the same shape, but 120° west and a bit farther south. If you jump ahead from the 2021 eclipse to the next one in its Triple Saros, you will find the total eclipse of 24 July 2055, 54 years later. Notice that this eclipse is in the same part of the world — crossing southern Africa — but farther south.
See how many other "partners" you can spot on the eclipse maps. You should notice that eclipses from the same series have the same "shape", and will be just slightly north or south of each other; but bear in mind the 120° westward shift (a third of the way round the world). You can check your findings against the eclipse list, which shows the Saros number for each eclipse. Also, each eclipse's data page contains links to the previous and next eclipses in its Saros series, and in the Triple Saros.
If you like, go on and read a slightly different view of the cycles behind the Saros.