The Fixed Signs are Taurus, Leo, Scorpio and Aquarius. As in the Cardinal and Mutable Qualities, each of them belongs to one of the four Elemental Triplicities. This is a matter of balance and Sacred Geometry.
When the Zodiac was designed, the ancients used geometry and whole numbers to represent "pure" forms in the universe. In this case, the Tetrahedron is an object of four sides, representing the elements. Each face is an equilateral triangle, representing the three qualities; this was an important meditation to the ancient sages. The Qualities were thought to impart their nature to the Signs, making the elemental characteristics more specific and more individualistic.
The Fixed Quality alters the pure nature of the Four Elements such that Fixed Signs express them in a more stable and persistent form. The four Fixed signs are generally more practical and more willful than signs belonging to the other two Quadratures. They are also more opinionated, and want to be right because they hate to admit it when they are wrong. They can be somewhat stubborn, and are more likely to fight for what they believe in than they are to change their beliefs. Fixed Sign people tend to build on the past, and are, on the whole, are more traditional that the other two groups of signs.
Taurus, being both an Earth Sign and a Fixed Sign, tends to be the most representative of the Fixed Quality. Although people born under Taurus are generally gentle and kind, they also can be very economical when it comes to the use of words. They are very slow to anger, but when they do become upset, the ferocity of their response can be measured on the Richter Scale, like any other earthquake. They are generally traditional in outlook, are patient in their labors and prefer the quiet of home and the countryside to the hustle and bustle of daily life.
Leo is the Fire Sign amongst the Fixed group. People under this sign are known for their warmth and generosity; they are the least impulsive of the Fire Signs. Leo captures the protective nature of the Fixed quality, in that they are usually more interested in preserving their relationships; they are not happy when they are on their own. A typical Leo will grab the center of attention wherever they are, much like the hearth was traditionally the center of focus within a home, or the fire was the center of a campsite or village. In other words, their fire has been "Fixed."
Scorpio is a Water Sign, and although emotional, the Fixed Quality enables them to stabilize themselves against the tides of emotion, hence they are the least insecure of the Water Signs. They are known for their emotional depth and their ability to see into the depths of others; this depth is part of the Fixed Quality. Some writers describe Scorpio as being like an iceberg (Fixed Water), because only a fraction of their inner nature is ever exposed to others. Scorpios like tangible evidence, like other Fixed Signs, and don't believe that justice is an abstract concept.
Aquarius is an Air Sign, and therefore the most mentally-oriented sign of the Fixed Quality. Although they may be innovative, surprising and seem to have their head in the clouds at times, the Fixed Quality makes this the most practical of the Air Signs. They can be uncompromising, and are determined to put their ideas into practice. Aquarians can be somewhat insensitive, especially if their ideas about the world are challenged; when they commit to something or someone, it usually is a binding proposition; as is typical of the Fixed Signs, Aquarius does things for keeps.
Because the Elements represent the fundamental "substance" of the signs, and the Qualities represent their characteristic actions, they are hard to separate from one another when studying sign-to-sign compatibility. The drive toward stability is so strong among the Fixed Signs that they tend to have trouble with people who like a lot of change in their life. As an example, the trustworthy and serious Scorpio will have difficulty understanding the changeable and mentally flexible Gemini, not just because they are different Qualities; both their Qualities and Elemental Natures are different. Scorpio does just fine with Cancer, despite the difference in quality, because the elemental natures are identical.
The difference in elemental nature in signs of the same Quality is usually moderated by that quality; for instance, Taurus admires the self-assuredness of Leo, and Scorpio is glad to take up the fight for Aquarius' cause. Simple elemental compatibility can be overcome by the Qualities; many astrologically unlikely couples depend on them.
Use this link to learn more about the Fixed Quality.