This group has been named the Fire Signs because these signs have a hot and dry nature, which are the fundamental properties of the Fire Element. Some of the common characteristics of this type are enthusiasm, impulsiveness and a fun-loving nature.
There is also a sense of adventure and opportunism that usually belongs to this group; people who are born into these signs, or whose horoscopes are dominated by them, are usually not averse to taking risks.
The Fire Signs were originally known as the Aries Trigon. The three signs belonging to this Trigon are Aries, Leo and Sagittarius. A Trigon is a group of three signs that are equidistant from one another when one measures around the circle of the Zodiac. In other words, if three lines are drawn to connect each sign in a Trigon to another they will form an equilateral triangle.
Like fire itself, people born with dominant fire signs will tend to jump to the nearest source of "fuel" (usually adventure or drama) and grow by consuming it. People born with the Element of Fire are known for their endearing warmth and their ability to "light up a room" with their presence. They are also known for their quick tempers; they tend to "flare up" quickly, but can cool off quickly and let the anger go.
Each of the Fire Signs has characteristics that they don't share with the other two. One of the reasons for this difference is because each member of the Trigon belongs to a different Quadrature or Quality.
Aries, being the Cardinal Sign of the group, wants to lead; the word "cardinal" means "leader." Because of this, an Aries person will find it easy to start things, but hard to finish them. They act on the basis of their passions without thought of the possible consequences.
Leo is the Fixed Sign of the group, and wants to rule. It's very rare for a true Leo to walk into a room without becoming the center of attention. It's harder for them to let go of anger than the other two.
Sagittarius is the Mutable Sign of the group, and is probably the most mobile of all signs of the Zodiac. The Sagittarian drive is to conquer, or to use a more modern term, to achieve; they usually have some difficulty with commitment, or staying in one place.
Because the triangle was considered the "first" stable form (you can't make a chair with two poles for legs that works very well), it was revered by ancient philosophers as being a "pure" form. Astrologically, signs that belong to the same Trigon work in harmony with one another. In other words, Aries should always be a good match for Leo or Sagittarius. But the Zodiac Signs alone are not all there is to astrology or astrological compatibility; due to other factors, such as the positions of the Moon and the planets at the time of birth, there are an almost infinite set of variations that can affect the strength of the elements.
In general, Aries, Leo and Sagittarius tend to get along well. Leo tends to be a little more self-conscious than Aries or Sagittarius, whereas Sagittarius tends to be more blunt, and more prone to errors in judgment. The Aries will often want to "win" so badly they'll overreach themselves. These are usually the characteristics of the Fire Signs that clash with one another.
All three signs are passionate about what they do, and are likely to underestimate or disregard danger, because they tend to be more strategic than tactical in their thinking. Most of the time, they do not mind taking risks, and this can often be an issue in fire-sign couples; they find it hard to save money, preferring to use it to enjoy themselves, and expecting it to somehow be there when they need it.
Astrological theory says that the Fire Signs are generally compatible with the Air Signs, because air is necessary to feed fire. According to the ancient definitions, the Fire Signs are hot and dry, while the Air Signs are hot and moist; they have the warmth in common and the moisture and dryness balance one another.
In other words, Aries, Leo and Gemini get along very well with Libra, Aquarius and Gemini. Again, this is a general idea based on the elemental theme of the Zodiac signs; sometimes Gemini might not like playing "court jester" to Leo, or Aries might not like to sit around and deliberate over the finer points of an issue with Libra.
In summary, the Fire Signs do well with one another, and have more in common with the Air Signs than the other two groups.