As mentioned the fire sign covers the Leo, Sagittarius, and Aries zodiac signs. They are all passionate, fiery, and energetic. They tend to be stubborn and hard headed which makes them one dimensional when they are passionate about something, in other words they will burn to possess that which they desire. They tend to be domineering and self centered, again, burning for that which they need.
The water elemental represents Cancer's, Scorpio's, and Pisces'. They are the exact opposite in many ways of the fire signs. They are deeply feeling and in touch with every thing around them, much like water often overwhelming and overlapping things and people they care for, or drowning them with passion. Like water, when overwhelmed by their surroundings they will often retreat deep within themselves like water in an underground cave.
The Earth elementals, Taurus, Virgo, and Capricorn's alike, are, well, very often down to earth, and very solid people like mountains. They are often steady and grounded. They are usually very rigid and set in their ways, but try and move a mountain one time. They also tend to be very inflexible and hard headed. Earth signs in general are what one might call the salt of the Earth.
Air signs of course are very ethereal, head in the cloud sorts. They are made up of the remaining signs of the zodiac, the Gemini, Libra's, and Aquarius'. Air signs, well, tend to be full of hot air. They are very social and can talk for hours, blowing in the breeze like the social butterflies that they are. They are very quick thinkers and hate to be tied down. Just like their elemental representative, they have to be able to blow freely in the breeze.
As you can tell, these are blanket generalities for three signs that don't fit every individual sign from every individual element. Elements help to associate certain groups of signs with other groups of signs. For instance, a fire sign is not likely to be a good emotional match with a water sign, as water puts out fire. However, an air sign is very likely to be a good fit for a fire sign, as air when heated by warmth tends to get excited and move even faster, and air feeds fire, making it grow, without air there can be no fire. These are the types of things the elemental signs can help you with in astrology.