So, what are equinoxes, why do they occur at varying times from year to year, and why is there not precisely 12 hours of daylight and 12 hours of night everywhere on our planet during the equinoxes?

     The seasons' starting times are governed by the Earth's motion around the Sun - or equivalently, from our point of view, the Sun's annual motion in Earth's sky.  The start of fall (for the Northern Hemisphere, for example, is defined as the moment when the Sun passes over Earth's equator heading south - a moment called the autumnal equinox. This moment can come at any time of day or night. 

     The Sun appears to move north and south in our sky during the year because of what some might consider an awkward misalignment of our planet. Earth's axis is tilted with respect to our orbit around the Sun. So when we're on one side of our orbit, the Northern Hemisphere is tipped Sunward and gets heated by more direct solar rays, making summer. When we're on the opposite side of our orbit, the Northern Hemisphere is tipped away from the Sun. The solar rays come in at a lower slant in our part of the world, which subsequently heats the ground less, making winter. 

     For a skywatcher on Earth (in the northern temperate latitudes), the effect is to make the Sun appear to move higher in the sky each day from December to June, and back down again from June to December. An equinox comes when the Sun is halfway through each journey. 

     This celestial arrangement makes several other noteworthy things happen on the day an equinox.

---Day and night are almost exactly the same length; the word "equinox"
comes from the Latin for "equal night." (There are two reasons why a look in your almanac will reveal that day and night are not exactly 12 hours long at the equinox. First, sunrise and sunset are defined as when the Sun's top edge crosses the horizon, not when the Sun's center crosses the horizon. Second, the Earth's atmosphere distorts the Sun's apparent position slightly when the Sun is very low.

 ---The Sun rises due east and sets due west. The fall and spring equinoxes
are the only times of the year when this happens.
---If you were standing on the equator, the Sun would pass exactly overhead in the middle of the day. If you were at the North Pole (during the autumnal equinox), the Sun would be skimming the horizon about to begin the six-month polar night.  (The opposite occurs after the vernal equinox in March.)