Why there is not an eclipse every month
If the Moon orbited in the same plane as Earth orbits the Sun, every new Moon would be a solar eclipse and every full Moon a lunar one. The orbit is tilted by just over five degrees, so at most new Moons the shadow passes above or below Earth, and at most full Moons the Moon passes above or below Earth's shadow. Five degrees is a small angle, but the Sun and the Moon are only about half a degree wide, so it is more than enough to miss by.
The two points where the tilted orbit crosses the plane of Earth's orbit are the lunar nodes, north and south. An eclipse requires a new or full Moon close to one of them. Close enough is about eighteen degrees for a solar eclipse and about twelve for a lunar one, with the exact limits deciding whether the result is total, partial or, for lunar eclipses, merely penumbral.
The season
The Sun passes each node once a year, and while it is within range every lunation that falls there is an eclipse. That window is about thirty-five days long, which is longer than one lunation and shorter than two, so a season always contains at least two eclipses — one solar and one lunar, a fortnight apart — and sometimes three. Two seasons fall in a year, roughly six months apart, which is why the dates on this site cluster and then leave long gaps.
The nodes themselves drift backwards through the zodiac, completing a circuit in about 18.6 years. Because of that drift the seasons come round every 173 days rather than every half calendar year, so they creep earlier by about three weeks annually, and the signs the eclipses fall in shift steadily too — a pair of signs holds the eclipses for around eighteen months before the nodes move on.
Kinds, and what decides them
A solar eclipse is total when the Moon is close enough to Earth to cover the Sun's disc, and annular when it is far enough away that a ring of sunlight remains. The two are the same alignment at different distances, and a hybrid eclipse is one that switches between them along its track because Earth's surface curves toward and away from the Moon. A partial eclipse is an alignment good enough to clip the disc but not to centre on it.
Lunar eclipses divide the same way. A total lunar eclipse has the Moon fully inside the umbra, Earth's dark inner shadow, and turns it a dull red because the only light reaching it has been bent through Earth's atmosphere. A partial has part of the disc in the umbra. A penumbral has the Moon in the outer shadow only, and the dimming is slight enough that most people looking up would not notice it.
The saros cycle
After 6,585.32 days — eighteen years, eleven days and eight hours — the Sun, the Moon and the nodes return to nearly the same relationship, and a very similar eclipse occurs. That interval is the saros, and the eclipses linked by it form a numbered family. The extra third of a day means each repetition lands about 120 degrees further west, so an eclipse family works its way round the planet in threes. A saros series runs for twelve to fifteen hundred years, drifting gradually across the globe before it expires.
Astrologically the eighteen-year return is the reason eclipse interpretation reaches for history: the practice is to look back eighteen years and nineteen years for what was happening at the same degree, rather than to predict forward. The specific part of an eclipse reading is narrow — a natal placement within a couple of degrees of the eclipse degree is contacted, and the rest of the chart is not.
Questions people ask about this
How many eclipses happen in a year?
Four in most years, and as many as seven when the seasons fall so that a third season clips the start or end of the year.
Why do solar and lunar eclipses come in pairs?
A season is longer than one lunation, so both the new Moon and the full Moon a fortnight away fall inside the window at the node.
What is the saros?
The 6,585-day interval after which a near-identical eclipse recurs, shifted about a third of the way round the globe.