The Cardinal Signs are Aries, Cancer, Libra and Capricorn. There is a representative of each Elemental Triplicity in this Quality, as is true for the other qualities. This is a fundamental observation from the standpoint of Sacred Geometry, in that the Platonic solid of the triangle, commonly called a Tetrahedron, is made up of four (Qualities) Equilateral Triangles (Elements).
The most important distinction between the Elements and the Qualities is that the Elements represent the fundamental nature of the Zodiac Signs, where the Qualities represent the temporary state of the Sun, Moon and planets while they are in that sign.
The characteristics imparted to the Zodiac Signs by the Cardinal Quality are mostly reflected by the forcefulness and the dynamic nature of the signs. They are signs that initiate beginnings, and generally put 100% of their effort into making their mark or getting ahead. They are all opportunists, and rarely miss the chance to "grab the brass ring", but are not very good at considering the "human element" or other's feelings, without a lot of influence of the Water Element, as in many people born under Cancer. People born under Cardinal Signs rarely wait for good things to happen to them; they will generally try to make things happen for them.
Aries, belonging to the Fire Signs, is always ready to start something, but often "burns out" before completing what they start. It's rare to find an Aries who isn't either striving to get ahead, whether in business or relationships. Because of the impulsive nature of Fire and the dynamic nature of the Cardinal Quality, they have the habit of jumping headlong into things; they're often too confident for their own good. An Aries is typically concerned with being a leader, thinking that it's natural for them to be first; this is also a Cardinal quality.
Cancer is a Water Sign, and is the most sensitive sign of the Cardinal group. Cancer's often become leaders because of their sensitivity toward the feelings and needs of others. They are usually very concerned with material security, because they believe that they like the power and service that wealth brings. Cancer also expresses its Cardinal Quality through moods that change with the regularity of the seasons. The intuitive nature of the Water Element, and the Cardinal opportunism combine to make Cancers very skilled business people.
Libra is an Air Sign, and is the most mentally oriented of the Cardinal Signs. Libras like to win, almost as much as Aries does. It's almost impossible to win an argument with a Libra because they possess the drive and opportunism of the Cardinal group, which allows them to seize on the smallest flaw and attack it with relentless logic. Because of this combination of Elemental Nature and Quality, however, they are not very good at seeing the human or emotional side of a question or decision that involves feelings.
Capricorn is an Earth Sign, and therefore the most practical of the Cardinal Signs. They are as obsessed with security as Cancer, but usually lack the intuitive sense of seeing opportunity when it arises, and tend to rely on work and tradition in order to become leaders. Capricorn makes a good leader because they rise to the top through effort, rather than through the desire to win. Capricorn does not like emotional display; they are far too practical and consider "scenes" as being a waste of perfectly good time. They are skilled, fair and practical in business.
Compatibility between the Qualities of the signs is not as important as that between the Elements, because the elements represent the fundamental nature of the signs, and the Qualities describe how those Elements take action. Because of the leadership drive of the Cardinal Signs, they may have problems with those Zodiac Signs that place individual freedom at the top of their agenda. This is part of why the social and logical Libra usually has a problem with the private and intuitive Fixed Sign Scorpio, or the ambitious and practical Capricorn may not find the ever-changing agenda of the Mutable Sign Gemini to their liking. In both these cases there is an elemental incompatibility; Libra gets along fine with the Fixed Air Sign Aquarius, and Capricorn rarely dislikes the company of Virgo, a Mutable Earth Sign.
These differences in element are often moderated between signs of the same quality; Aries appreciates the nurturing and care they feel in Cancer, and Capricorn is the only sign who can win an argument with Libra. Unless there are extremely difficult patterns among the planets in two individual birth-charts, the Quality can override simple elemental incompatibility.
Use this link to learn more about the Cardinal Quality.