Why that Figure-Eight Shape?

You probably already know that the sun's apparent daily path across the sky is different at different times of year.

During Northern Hemisphere summer it's farther north than at other times, while during Northern Hemisphere winter it takes a more southerly route.

You may not be aware that its east-west position also varies. If you compare the time shown on a sundial against a clock (ignoring Daylight Saving) you will find that during some parts of the year the sun runs a few minutes ahead of schedule while at other times of year it lags a few minutes behind.

This arises from the geometry of the situation. One simple way to look at it is from the astrologer's geocentric view of the sun making its yearly journey from west to east around the Zodiac while the earth spins on its axis under it. Since the sun will have moved a tiny bit (about one degree) in the time it takes the earth to rotate once, a solar day is actually slightly longer (about four minutes) than one complete rotation of the earth. That is why any given star rises and sets about four minutes earlier every night when you measure it by sun time.

Now since the earth is roughly spherical, the parts of it closest to the equator are moving faster than the parts near the poles. Or, to put it another way, time zones get narrower as you move away from the equator. Near the Solstices, when the sun is farther from the equator, the earth has to turn farther to catch up with it than when it's near the equator around the time of the Equinoxes.

Thus right at the Solstices, when the lengths of the days and nights are barely changing, you will see both sunrise and sunset getting a little bit later day by day. In other words, the date of the earliest winter sunset will not be the same as the date of the latest winter sunrise. Likewise, the dates of earliest and latest summer sunrise and sunset will be different.

If you plot the position of the sun at the same clock time every day (ignoring Daylight Saving) for a year you'll get something that looks like a figure eight. It's a little bit lopsided because Earth's orbit is not a perfect circle, but that's the general idea.

The smaller loop of the figure eight is toward the north, where the sun is at the time of the June Solstice. The larger loop is toward the south, the December Solstice. In both cases, the sun is moving from west to east (falling behind clock time) as it rounds the turns at the ends of the figure.

In almanacs and such, the number labeled "Declination" is how far north or south of the Equator the sun appears to be (negative numbers are usually south of the equator if not otherwise specified). The "Equation of Time" is how early or late sun time is compared to clock time (which is an average over the course of the year).

Here's what Wikipedia has to say about analemmas.