The year is naturally divided into four parts by the two equinoxes (when the sunrise is the closest to exactly due East, and the sunset is similarly due West) and the two solstices (when the sunrise and sunset are the furthest North or South that they will be that year). These four quarters are the basis of what we often now call the four seasons.
It seems natural also to be aware of the midpoints of these quarters - that is, half-way between a solstice and the following or preceding equinox. These four “eighth days”, sometimes called the “cross-quarter” days, are the traditional Celtic festivals of Beltaine (rites of Spring), Lugnasagh or Lammas (first fruits of harvest), Samhain (all souls) and Imbolc (mid-winter). Seasonal festivals might have been celebrated on the full moon - or for some festivals, the new moon - following, or perhaps closest to, these points in the solar year. The date of Easter is still found like this, being based on the first full moon after the Spring equinox. The Chinese new year is generally on the New Moon closest to Imbolc!
The modern European or Western calendar - the “Gregorian” calendar - divides the year into twelve rather than eight, this may be inspired partly by the fact that there are approximately twelve lunar cycles in a year (there are 12 synodic months plus about ten or eleven days, in a year), or possibly just by the fact that twelve is a really neat number to divide a circle into anyway! - see "the Rule of Twelfths" below.
The Gregorian calendar doesn't epecially note the solstices or equinoxes, and is based at apparently a rather arbitrary point in the solar year. (In fact January 1st is currently close to, but not exactly at, Earth’s “perihelion”, the moment when the Earth is closest to the Sun in its annual elliptical orbit, but that itself is precessing slowly, and anyway has no major effect on the terrestrial seasons.) Nevertheless, historically the application of leap-years has been carefully regulated to ensure that the calendar does remain synchronised with the solar year.
This is a “rule of thumb” traditionally used to estimate the progress of the ocean tides, though it's really a general observation on the rate-of-change of any quantity which varies approximately "sinusoidally", like a sine wave. One characteristic of this kind of movement is that it moves fastest in the middle, and slows down as it approaches the turn-around. (A definitive example of sinusoidal motion is the back-and-forth motion of a point that's travelling at a steady speed around a circle, but we only look at the movement along one axis - like looking at a record deck from the side, with one eye shut. Mathematically, the rule of twelfths is equivalent to noting that sin(30°) equals ½ exactly, and sin(60°) is roughly equal to ⅚.)The rule-of-twelfths considers both the overall range of motion, and the total time taken for one cycle, to be each divided into twelve equal parts. In the case of tides, from high to low and back to high again takes about 12 hours, so that makes the reckoning easier. (Which is a good reason in itself to divide the day into 24 equal parts!) In the table below, the tide height is shown as varying from +6 to -6, so the actual tide height either side of the mean is proportional to this. Starting from the high tide, during the first twelfth of the time (nearly one hour), the height of the tide goes down by one twelfth of its range. During the second "hour", it goes faster, and drops by two-twelfths of its range; during the third time-interval, it goes faster still, and decreases by another three-twelfths of the total range. By now, it's half-way down (the height is shown as zero), and the second part is a kind of mirror of the first, getting slower as it reaches the minimum (low tide). Then it comes in again, in much the same way.
Because the Earth is spinning, stars appear to rise over the Eastern half of the horizon, move across the sky, and set in the Western half. (Although some stars - those whose latitude is greater than the observer’s - either never set, or never rise in the first place. This depends on where the observer is on Earth. The ones that never set are called “circumpolar”.) The Sun of course also rises and sets like this, but takes slightly longer than the stars to do so because the Earth’s orbit round the Sun makes the Sun appear to move slowly Eastwards against the background of the stars, during the year. A sidereal (fixed-stars-based) day is about 4 minutes shorter than a solar day.
If you watch the Eastern half of the sky during the early hours of the morning, while it’s still dark, you can see stars rising. Eventually dawn comes, and you can no longer see the stars - though they’re still there, the sky is too bright to see them. The following night, you’ll see the same stars about four minutes earlier (because the sidereal day is shorter than the solar day.) So you may see a star rise, just before the Sun, that you couldn’t see at the end of the previous night because it was hidden by the Sun rising first. This is called the heliacal rising of that star. For a given star and a given place, (assuming consistently clear weather!) it happens once a year. For example, this annual first sighting of Sirius rising was used as a marker for the passage of the year by the ancient Egyptians.
Over the following months, the star will become more visible, and visible earlier and earlier in the night, until it finally becomes an evening star, and sets just after the Sun - heliacal setting - and then cannot be seen again until it is again a morning star, months later. The word heliacal derives from “helios”, the Greek name for the Sun.
"Co-ordinates", in this context anyway, means numbers which are used to specify a position. On the celestial sphere, just as on the surface of the Earth, we need a pair of numbers. One meaures around an equator or similar; the other number measures up or down (plus or minus) up to 90 degrees from this, to cover the whole sphere.
There are three main spherical coordinate systems used in Earth-based astronomy: Ecliptic Longitude and Latitude (based on the Earth’s orbit), the equator-based “Right Ascension” and “Declination”, and an observer’s horizon-based “Azimuth” and “Altitude”.
To specify a point on the Earth’s surface, we can use “latitude and longitude”, based on the Earth’s equator, with a “zero” point on the Greenwich meridian. Longitude is measured East or West from this meridian, and latitude up to 90 degrees North or South from the equator. We can project this system out to the celestial sphere, but for the zero point we then use “Aries zero” - the position of the Sun on the Spring equinox. This produces the celestial coordinate system called “Right Ascension and Declination” - R.A. is like longitude, and declination is the celestial latitude. Right Ascension is conventionally measured Eastwards - the same direction as the Sun appears to travel against the background of the “fixed stars”, during the year. Also, it’s often measured using “hours” as units - 24 hours being equal to 360 degrees in this context. Most star tables give positions using this system; it’s fairly easy to figure out where to find a star in the sky at a given time, from this information.
We can instead use the ecliptic - also with a starting point at Aries zero - to provide a basis for our coordinates. In this case, we’re measuring around the zodiac - the apparent path of the Sun through the stars - which is the way many tables of planetary positions are presented. Ecliptic longitude, like R.A., is measured from West to East. Converting accurately from ecliptic to equatorial coordinates will require some sort of (spherical) trigonometrical calculations, or the equivalent. (For example, in the SkyScape program, positions on the sphere are sometime stored as 3-dimensional “vectors” (of unit length), and the coordinate system is then changed by matrix multiplications on those vectors.)
Finally, for some specific location on Earth, we could base our coordinates on the sky and horizon that would be experienced there; measuring a star’s altitude above that horizon, and its “bearing” or direction around the horizon - conventionally measured Eastwards from North - which is called its azimuth. This pair of numbers - Azimuth and Altitude - can be called horizon-based, or horizontal, coordinates.
This rather grand-sounding name is used to describe the way in which the time on a clock - which presumably measures the passing of time in a steady, uniform way - differs from the time on a sundial. A perfectly-adjusted sundial will show the celestial longitude of the Sun, relative to the longitude on Earth where the sundial is. When the Sun is highest in the sky, and the shadow of a vertical stick in flat ground points due North (or South in the Southern hemisphere), then the sundial will tell us that it’s midday, 12 noon, in “local solar time”. However, the interval measured by an accurate clock, from one such “solar noon” to the next, actually varies, slightly, over the course of a year, sometimes being a little more and sometimes a little less than 24 hours on the clock.
There are two things which cause this: firstly the fact that the Earth’s orbit around the Sun is not exactly circular, so the angular speed of the Earth around the Sun is not exactly constant; and secondly the fact that the axis of spin of the Earth is not at right angles to the plane of its orbit. These facts cause the apparent speed of the sun through the sky as it rises and sets, to be not exactly constant from day to day. (It is very nearly constant, as the main reason for the rising and setting is the spinning of the Earth, which does continue at a very steady rate, although even this is slowing down very gradually.) Then each day at the moment when the sundial shows midday, we can look on the clock to see how much the time shown there is before or after 12:00, and over the course of a year, this will give us a graph of the equation of time. The usual convention is to represent this value as positive when the sundial is ahead of the clock, but this not always followed. (Note: the clock needs to be set to the local mean time zone of the sundial’s location! E.g. have the sundial at Greenwich and set the clock to GMT.)
The sundial can be ahead (fast) by as much as 16min 33sec (around November 3) or behind (slow) by as much as 14min 06sec (around February 12). The equation of time has zeros near 15 April, 13 June, 1 September and 25 December, when sundial time and clock time agree. There is more explanation, and some images of the graph, on wikipedia.