- People ask me about this stuff sometimes, so I thought I'd write some of it down. I hope it helps..
Everything is going round everything else, but bigger things tend to stay relatively still. The Sun, being by far the heaviest thing in its vicinity (our 'solar system'), is more or less at the centre of mass of the whole system. The planets, including Earth, continue to orbit the sun, at various distances and speeds. Most planets have smaller things ('moons') orbiting them.
Earth is a bit unusual within our solar system, in that it only has the one Moon, and that this Moon is larger than many of the other planets' moons. However, the average density of the whole Earth is almost double the density of the Moon - the Moon's iron core is relatively much smaller than Earth's - which makes Earth a good eighty times more massive than its Moon, so the centre-of-mass of the Earth-Moon pair (which is the point around which they really both orbit) actually is within the Earth (about 1700km down).
So, we think of the Moon orbiting the Earth, and the Earth (or the Earth-Moon pair) orbiting the Sun. This orbit, of the Earth around the Sun, is approximately circular, with the Sun at the middle. A better approximation is that it's very slightly oval, with the Sun not quite at the centre, but in any case, like a circle, it's flat, i.e. in one 'plane'. The centre of mass of the Earth+Moon system doesn't leave this plane as it orbits the Sun, though the Moon itself is not always in this plane. This plane is called the plane of the ecliptic, or the ecliptic plane. It is called this because eclipses can happen only when the Moon is very close to this plane at the time of New or Full Moon.
As well as moving around the Sun in its orbit, the Earth is also slowly spinning 'on its axis'. This spinning is why an observer on Earth sees the Sun and stars rising and settting daily. The Earth spins around once a day (approximately) - though because of its size, that translates to a speed at the equator of over 1600km per hour! Its axis of spin is pretty constant - the straight line through the middle of the Earth joining the North and South poles. As the Earth spins, the two 'poles' stay in the same place - I mean they keep the same stars overhead, while spinning underneath them - but other points on the planet move around the North-South axis once per 'day' (that's a 'sidereal' day to be precise), causing the stars (and Sun and Moon etc.) overhead to apparently move from East to West. If you're on Earth, and away from the poles, then in a clear sky you'll see things rising (along the Eastern half of your horizon) and setting (along the Western half). If you were in space above the North pole, the Earth would be appear to be spinning anti-clockwise below you.
If we look at the clear night sky, the stars appear to lie on a huge sphere surrounding us. Even the Moon, which is really much closer, appears to be just as unreachable, on that 'inverted bowl we call the sky'. When talking about the apparent movements of celestial bodies as seen from Earth, we often want just to consider in what direction we must look to see them - and geometrically, this is the same as imagining that they all lie on an infinite 'celestial sphere', with us at the centre.
The (imaginary) line around the Earth, half way between the poles, is called the equator, and lines on the Earth parallel to the equator are called 'circles of latitude'. These circles can all be imagined to be projected out onto the celestial sphere, to help us describe what we see. Because the Earth is spinning, an observer on Earth (who is not too near the North or South poles) will see the Sun, Moon, stars, and sometimes other planets, rising and setting - and in between rising and setting they move across the sky along celestial circles of latitude, parallel to the celestial equator.
As it happens, the plane of the equator (that is, a plane perpendicular to the axis of spin of the Earth) is tilted, in a fairly unchanging direction, and at a pretty constant angle - about 23½ degrees - relative to the plane of the ecliptic. As Earth orbits the Sun, this tilt causes the seasonal changes in day-length during the year.
The Sun appears North of the equator during half the year - and then South the other half of the time - because the Earth's axis keeps pointing in the same direction, while Earth itself is orbiting the Sun. So one day, the North end of the Earth's axis may be pointing somewhat towards the Sun; but then six months later it will be pointing away from the Sun. So the Sun's declination (its angular distance from the equator, as seen from Earth) varies from about 23½ degrees South to 23½ degrees North, during the year. The equinoxes are the times when this declination is zero (i.e. the Sun is on the celestial equator, i.e. directly overhead somewhere on Earth's equator). The times of maximum and minimum are called 'solstices' - meaning that the Sun (its declination anyway) is 'stationary'.
The equinoxes occur at the two points in the Earth's orbit when the Earth's axis is at right angles to the line from Sun to Earth. The day-length then (around March 21st and September 22nd) is almost exactly 12 hours, everywhere on Earth. The solstices occur when the angle between the Earth's axis and the Sun-Earth line is a minimum or maximum, around June 21st and December 21st.
Projected onto the celestial sphere, the ecliptic plane becomes a 'great circle' - a line going right around the sphere, making a circle whose centre is at the centre of the sphere. In other words, seen from Earth, anything on the ecliptic plane will appear to be on a certain line on the celestial sphere - a great circle like the equator, but tilted to it at about 23½ degrees. The band of constellations around this line is called the zodiac. (The etymology is a bit uncertain; this word may derive from the idea that these constellations mostly represent various animals, or it may possibly be derived from a Sanskrit word for the 'path' of the Sun and planets).
As it happens, all the other planets orbit the Sun in planes which are not far off our ecliptic plane. Our Moon also - unusually for solar system moons in general - is orbiting in a plane tilted only a few degrees from the ecliptic. This means that, seen from Earth, all the planets, and the Moon and the Sun, are only ever seen in directions not far from the ecliptic - i.e. within the constellations of the zodiac.
At the Northern hemisphere Spring equinox (about March 21st), the direction of the Sun from Earth is called, for historical reasons, 'the first point of Aries' or 'Aries zero'. This point is used as the origin or zero point for the definition of celestial longitude (measured around the ecliptic) and Right Ascension (around the equator).
The term 'Aries zero' is still used, although this point is no longer in the actual constellation of Aries. The direction of this 'equinox point' - which is one of the two points of intersection of the ecliptic and the equator on the celestial sphere - slowly moves around the ecliptic (in the opposite direction to the Earth's orbit) in an approximately 26000-year-long cycle, as the axis of spin of the whole spinning Earth slowly changes direction. The Earth still spins about the same axis within itself; the North and South poles don't move on the Earth's surface, but the whole thing slowly rotates, relative to the "fixed stars" surrounding our solar system (and clockwise, looking from the North side of the ecliptic plane), around an axis perpendicular to the ecliptic.
So what we now call the "pole star" does not remain directly over the North pole; the Earth 'precesses', like a spinning-top or toy gyroscope can do (but much more slowly). This is mainly caused by the Sun's gravity on the equatorial bulge of the Earth, tending to make the axis more at a right-angle to the ecliptic; this gradual movement of the equinox points is called 'precession of the equinoxes'.
This movement is why, about 2000 years ago, Aries zero was at the start of the constellation of Aries, but now it's moved (backwards) through most of Pisces, and will enter Aquarius soon. (That will be the 'dawn of the age of Aquarius'. The exact moment when this will happen depends on where you draw the exact boundaries of the star constellations.)
Using the Zodiac names like this (to measure around the ecliptic from the Spring equinox point) is called the tropical zodiac. (The constellations of stars themselves may be called the sidereal zodiac.) The word tropic, from the Latin for 'turn', relates to the solstices, where the Sun 'turns' from moving Northward to moving Southward, or vice versa. The 'tropic of Cancer' is the circle of latitude on the celestial sphere that the Sun just reaches on the June solstice, when it is at the 'first point of Cancer' in the tropical zodiac. Capricorn is similar, South of the equator, for the December solstice. ('Tropics' can also refer to the two circles of latitude on Earth, at 23½° North and 23½° South, where the Sun is just overhead at noon on the solstices; or to the area of the Earth around the equator, between these two circles, where the Sun can sometimes be right overhead.)
The point where the Moon in its orbit around the Earth crosses the ecliptic plane from South to North, is called the Moon's 'Ascending Node'. (The 'Descending Node' is in exactly the opposite direction; being where the Moon's orbit crosses the ecliptic as the Moon goes from the North side of the ecliptic to the South.) Because the Moon goes round the Earth approximately once a month, it crosses the ecliptic twice in that time, once 'ascending', once 'decending'. However, the exact direction of the tilt of the Moon's orbit slowly varies, in other words the direction of the node is not fixed, but circles the Earth (incidentally in the opposite direction to the orbit of the Moon itself!) once every 18.6 years approximately.
This movement of the Moon's nodes is a result of the Sun's gravity trying to pull the orbital plane of the moon towards the plane of the ecliptic. Combined with the momentum of the Moon in its orbit, this results in a rotational effect similar to the precession of the Earth on its axis, so we can call it the precession of the Moon's nodes.
Because the Moon's nodes are slowly moving backwards around the zodiac, a 'Draconic month' - which is the time for the Moon to travel in its orbit around the Earth from the ascending node, all the way around to meet the ascending node again (which itself has moved backwards slightly in the meantime!) - will be slightly shorter (by just over 2½ hours) than the time it takes to get round to its starting position relative to the background stars (a sidereal month). The time from one new moon to the next (a Synodic month, or colloquially just 'lunar month'), however, is a bit more than 2 days longer than that, since the Moon has to travel a bit further to keep up with the changed direction of the Sun, as the Earth-Moon system orbits the Sun in the meantime.
Remember that rising and setting of objects in the sky is caused by the Earth's spinning, and so the apparent movement of any of these objects across the sky is parallel to the Earth's equator. The distance the object maintains from the equator is called its 'declination', and will determine, for any given latitude on Earth, exactly how far North or South (of due East and West) it will rise and set.
It's often easier to see the sun set than to see it rise - it's often more misty in the morning! However, whether we can see the Moon rising or setting depends also on its phase: if the Moon is fairly new, it will set soon after the sun, so although its setting can be seen, its rising will be in the morning after the Sun, and hard to observe, because it will be in daylight! The full Moon can often be observed just as easily rising or setting, but once it is well on the wane it sets after the sun has come up, and it can best be observed rising, between midnight and dawn. Stone circles are thought to be places where the rising or setting of the Sun, Moon or stars could be accurately observed in ancient times. To observe the declination of an object, it's enough to see it either rise or set. Thus the Moon's changes in declination can be tracked through the month by watching where it sets, during its waxing phases, and where it rises, during its wane.
Eclipses (solar or lunar) occur when the Earth, Moon and Sun are in an almost exact straight line. They're approximately in a straight line every new and full Moon, but because the plane of the Moon's orbit around the Earth is slightly tilted relative to the plane of the Earth's orbit around the Sun, they don't usually line up exactly enough to make an eclipse.
The plane of Earth's equator is tilted at about 23½ degrees to the plane of Earth's orbit around the Sun (the 'ecliptic'), and is always tilted in the same direction (this direction actually does change, but very very slowly, with the precession of the equinoxes, in an approximately 26000-year-long cycle). The plane of the Moon's orbit around the Earth is also tilted at a fairly constant angle (just over 5 degrees) to the ecliptic plane, but the direction of this tilt varies in a 18.6-year-long cycle. When the two tilts are combined, the result is that the plane of the Moon's orbit is tilted relative to the plane of Earth's equator, by an amount which varies from about 18 to about 29 degrees (and back again) over 18.6 years.
This variation can be directly observed from Earth, by noting exactly where the Moon rises or sets, day after day. Just as the Sun rises and sets further North in June, and further South in December, so the Moon rises and sets further North or South as it orbits the Earth during a month. The furthest North or South it gets during a month, is a measure of the combined tilts described above, and so this measurement itself has minimum and maximum values, which occur in an 18.6-year cycle. By observing where we are in this cycle, it becomes possible to some extent to predict eclipses.
The Moon's declination varies from North to South and back again during a month - though the amount it varies (North and South) during that month can be as much as 29 degrees or as little as 18, as decribed above. The moments when the Moon's declination is at any such minimum or maximum is called a 'lunistice' (like 'solstice', but lunar). The exact declination of the Moon at these lunistices does vary, from about 18 to 29 degrees (North and South), and back again, over a 18.6-year cycle, and observing this cycle can be used to predict eclipses. The time when the lunistices are at their minimum value is called a 'minor standstill', and the time when they're at their maximum is called a 'major standstill'. The lunistices happen when the Moon is close to zero Cancer (North) or zero Capricorn (South).
So the minor standstill is when the tilt of the Moon's orbit is in the same direction as the tilt of the Earth's equator - so the Moon as seen from Earth will never be very far from the equator. Just over nine years later, the tilts are in opposite directions, so the Moon's declination (angular distance of Moon from the equator) can get biggest. In both cases, the Moon's nodes are lined up with the intersection of the Earth's orbit and the Earth's equator - in other words the directions Aries zero and Libra zero. So if the Sun is also near Aries zero or Libra zero - in other words if the current date is close to either of the equinoxes - then we may see an eclipse or two that month.
Half-way between minor and major standstills, the eclipse seasons will be around the solstices (summer and winter). The eclipse season moves backwards through the year (as the Moon's nodes move retrograde through the zodiac), going round 12 months in about 18.6 years. So if one year there is an eclipse of some sort in early February, say, then there will likely also be eclipses around the beginning of August (six months later), and the next year we can expect eclipses to possibly occur in late January and mid July; they won't occur far from these times. There will usually be a solar eclipse every five-and-a-half to six months, and they are usually accompanied by one or two lunar eclipses (on one or both of the full moons adjacent to the solar eclipse new moon). Occasionally, there will be solar eclipses on two consecutive new moons; these are then usually partial eclipses, and surround a total lunar eclipse on the full Moon.
At a major standstill, the Moon's ascending node is close to Aries zero (and moving into Pisces); at a minor standstill it's near Libra zero. The actual moment of the greatest declination North or South, is the 'major standstill' itself - once for North and once for South. These are usually a fortnight apart, but not always. Interestingly these standstills seem always to happen around an equinox, apparently because the amount of inclination of the Moon's orbit itself has a slight variation, with a 6-month period, as the Sun's gravity tends to pull the Moon toward the ecliptic plane. This results in the orbital tilt itself being greatest when the Moons nodes are in line with the Earth-Sun line. Nevertheless, the Moon's greatest declination (North and South, each month) changes very little over many months. The important point is that in the year of a major or minor standstill, the eclipses will occur around the equinoxes (March and September). By continuing to observe the Moon's changing declination, by seeing where it rises or sets, we can keep track of where we are in this cycle.