The Saiyans were one of two races who lived on Planet Vegita. The other race was the Tsufuru. As the Saiyans were stronger than the Tsufuru, the Saiyans destroyed all of the Tsufuru. One thing that sets the Saiyans apart is their tail. If they look at the light from a full moon they can turn into Oozaru, a giant ape. In this form they are 10 times stronger than normal.
After conquering the planet the Saiyans were hired by an alien named Frieza. The Saiyans' mission was to conquer worlds and then sell them to the highest bidder. However, Frieza became worried about the Saiyans' power and destroyed planet Vegita. The only Saiyans left were Goku, Vegita, Brolli, Nappa, Raditsu, Paragus, and Taurus. During the course of DragonBall Z, the only true Saiyans remaining are Vegita and Goku.
The Saiyans are a warrior race and as such have incredible fighting abilities. As they fight their skills increase but unlike most races the Saiyans get stronger after recovering from a severe battle. After a certain point the Saiyans no longer gain greatly in strength. After they reach this plateau they have the ability to reach a new level of power. The Saiyans have the ability to achieve Super Saiyan Stage 1. However there are some conditions to be met. The person must have a pure heart, be strong enough to surpass a normal Saiyan. With these met all that is needed is extreme rage to trigger the transformation. Once transformation is accomplished it is easier to accomplish the next time do the fact that the rage is not needed again. However after achieving Super Saiyan Stage 1, the Saiyan is capable of attaining even more strength. There are a total of five levels. There is Super Saiyan Stage 1, Ultra Super Saiyan, Super Saiyan Stage 2, Super Saiyan Stage 3, and Super Saiyan Stage 4. Of the two remaining true saiyans Goku reaches all the levels and Vegita reaches SSJ1, USSJ, SSJ2, and SSJ4. Sidenote: If the Saiyans still have their tail they can reach the level of Super Saiyan Oozaru
On the Earth Goku and Vegita marry and have children with Saiyan traits. Goku and his wife have to childern. One is Gohan and the other is Goten. Vegita and his wife have two children, one is named Trunks and the other is Bra.
After Goku and Vegita there are no more true Saiyans and the race lives on in the bloodlines of their children and grandchildren.