The Elemental Trigons of the Zodiac are fairly well known, in that almost all Sun-Sign booklets and beginner's astrological literature has some reference to the Fire, Water, Air and Earth Signs, especially when they discuss compatibility.
Less is known about the sign Qualities, or Quadratures. Although they are fundamental to the construction of the Zodiac, their role in the nature of the signs and compatibility between people of different signs is not as obvious. Let's first explain what the Qualities are.
From a purely mechanical point of view, Zodiac is a coordinate system for measuring the apparent path of the Sun through the sky. During the year, the Sun moves North or South of the celestial equator, which is the Equator of Earth expanded into space.
In both hemispheres, the Sun's position, when it crosses the equator from South to North in reference to the equator is called, by astronomers and astrologers alike, Aries. The date on which is does so is called the equinox in both hemispheres, and in the Northern Hemisphere, it is the Spring Equinox.
Referring to the northern hemisphere, it continues to rise a little bit further north every day until it reaches its northernmost point; this means that every day it takes a little longer to move from Sunrise to Sunset. When it reaches this point, which is known by astrologers as Cancer, it's the longest day of the year. Maps and globes call this latitude (the measure of distance from the equator to the poles) the Tropic of Cancer.
It then begins to rise and set a little further south every day until it crosses the equator going south, at the Autumn Equinox, or Libra (many star maps use the Zodiac symbols to mark these points) as the days get shorter until it reaches its southernmost point, which is the Winter Solstice, or Capricorn. The latitude at which the Sun would be directly overhead when this happens is called the Tropic of Capricorn on maps. These four signs are called the Cardinal Signs.
It's often commented that Cardinal means "leader" but the old astrologers used to use the term "moveable" or "pivotal" signs because they are the points at which something significant happens that changes the length of day and night. It was said in old texts that the "air" changes on these days; they are the beginnings of the four seasons.
As the Sun moves from one Cardinal point to another, it passes through the "Fixed" Signs, Taurus, Leo, Scorpio and Aquarius. These are stabilizing influences that are more typical of the season that is active during the Sun's course than the other two signs that make up the quarter of the year. As you will see in the articles about the Fixed Signs, the characteristics of people born during their influence tend towards stability and persistence.
After leaving the Fixed Signs, the Sun passes through transitional zones in which one season gradually gives way to the next; these are called the "Mutable Signs." Mutable means changeable, but in ancient times, and in India to this day, they are called "dual", "common" or "double-bodied" signs, because they sit between two signs of different quality and partake of them both. They represent the end of one phase of the Sun's Path and the beginning of another. People born under Mutable signs are known for their flexibility and communicative natures.
What's most interesting about the qualities is how they change the nature of signs in the same Element. For instance, the Cardinal Signs of a given element are the most ambitious in terms of carrying out the agenda of the element. Aries, being the Cardinal Fire Sign, is the most ardent innovator among the Fire Signs, and always ready to start something new. Scorpio, being a Fixed Sign, is the most stable of the Water Signs, and quite capable of controlling their emotions in order to gain access to their insight into themselves and others. Virgo, being a Mutable sign, is the most communicative of the Earth Signs, always ready to share their accumulated knowledge and experience with others. Although astrological compatibility is not directly related to the Qualities of the signs, it is a most powerful factor in moderating the Elemental natures, and can make for common ground between signs that are of supposedly incompatible elements.
For instance, Gemini and Virgo have a tendency to get along very well, although they are of "clashing" elemental natures. They do, however, share the same quality and the same planetary ruler. They may annoy each other from time to time, but they are a terrific team at work or play.