History of the Saiyans ¹

The Saiyans are the fictional race from the Planet Vegeta. Goku is one of them. The Saiyans are an aggressive warrior race that live to fight and are born with a monkey's tail with is the only thing that differentiates their appearence from that of humans.
On a full moon, their tails cause the Saiyans to become Oozarou, namely 50ft were-apes, that releases the Saiyans' full potential. Energy and strength-wise. The drawback is, however, that once transformed, the Saiyans have no control over their enormous power and run rampant. The only Saiyans that are still able to have full control are the Elite: kings, princes and generals.
The tail is also the Saiyans' weakness. If grabbed (in their human form), it forces the Saiyan to fall to their knees in pain. In ape-form it causes the Saiyans to feel the pain but get angry from it.
To defeat a Saiyan's ape-form, the tail must be cut or the moon must be clouded for them to revert back to their human-like appearence.
Once again, the Saiyan Elite have evolved past their tails as a weakness.

The Saiyans worked as space pirates for alien species, using their aggression to destroy the inhabitants of planets ready for the potential buyers.
When there is a planet that is deemed weak by the Saiyans², they send Saiyan babies through space to those planets, and when the moon is up they will kill everything.
Since not all planets have moons, the one-man saiyan spacepod will create a projection.
Stronger planets require a few Saiyan warriors to fight.

Goku was sent to Earth as one of these babies. Found by an old man named Gohan who adopted him, Gohan could not control Goku's anger until the baby fell into a ravine and hit his head. The bump made Goku appreciate all life. Unfortunately it was Goku's tail that would kill his adopted grandfather and set up the events in Dragonball though Goku had always believed his grandfather died at the hands of the 'monster that comes out at night'. Before Dragonball Z, Goku's tail was cut off permenantly.

Even without a tail, Saiyans have bizarre attributes that make them potentionally the most powerful race in the universe.
Saiyans get stronger if they lose a fight. If they lose, their power doubles. They love to fight and, without their tails they usually have to be destroyed to stay down³.

The Saiyans believed in the legend of the 'Super Saiyan' as did tyrant Frieza, knowing that he would be destroyed by a Saiyan warrior he purged the universe of all but four saiyans who weren't on Planet Vegeta at the time Frieza blew it up.
These saiyans were: Prince Vegeta, Nappa, Raditz and Goku.

The legend was that once every 1000 years a Super Saiyan would be born.
In climax of the Frieza Saga, it was Goku who transformed into the Super Saiyan to beat Frieza. But as the series continued, the ability to become a Super Saiyan was more 'scientific'.
When Goku's rage became so high he changed into a Super Saiyan. From then on he could change whenever he 'powered up'. This would later be the same for the Saiyan prince, Vegeta.