In the beginning the Saiya-Jins shared their planet with the technically advanced but weak Tsufuru-Jin, on the planet then known as the Planet Plant. The Saiya-Jins were much stronger then the Tsufuru-Jins but they were kept in check but the advanced technology of the Tsufuru-Jins. But one night something happened that only happened once every 8 years, there was a full moon. This allowed the Saiya-Jins to undergo the "Saiya-Jin Transformation", as Vegeta puts it, or changing to Oozaru form (see below). From then on the poor Tsufuru-Jins had no chance of defeating the Saiya-Jins, the planet was renamed the Planet Vegeta after the king, Vegeta Ou.
From then on the Saiya-Jins adopted the Tsufuru-Jin technology, including the space pods, which allowed contact with other civilizations. This led to their meeting with the evil space tyrant Freeza, who used the Saiya-Jins as mercenaries, wiping out the inhabitants of any planet he desired. Eventually the Saiya-Jins became tired of working for Freeza and with Vegeta Ou Leading them they revolted to save the young prince, the Vegeta we know and love. Unfortunately this rebellion was swiftly defeated by Freeza, who then proceeded to destroy the Planet Vegeta to stop any more rebellions. The only Saiya-Jins who survived were Goku, Prince Vegeta, Raditz, Nappa, Taurus, Paragus and Brolly.
All full-blooded Saiya-Jins share several common features. They all are born with the hairstyle they keep through out life, all are very strong and muscular and they all have a tail. The tail is somewhat of a weakness for lesser Saiya-Jins, as any pressure applied to it is temporally paralyzing, but stronger Saiya-Jins, who have also strengthened their tail, are not affected at all by this. The tail is also necessary for the transformation to the Oozaru state.
Achieved by: All Saiya-Jins
The first of the SSJ transformations. This form is characterized by a massive leap in power, golden hair and green eyes. The first one to reach this stage was The Legendary Super Saiya-Jin, the only one that Freeza ever feared, but the first we see in the series is Goku when fighting none other than Freeza. The transformation is triggered when someone of pure heart (Good or Bad) experiences extreme emotion, this takes over and allows the change. With training, afterwards the transformation is controlled and happens at will.
Achieved by: Brolly, Goku, Vegeta, Mirai Trunks, Gohan, Chibi Trunks, Goten, Vegetto and Gogeta
Ultra Super Saiya-Jin
This is a midpoint between SSJ and SSJ2. At this stage a Saiya-Jin's muscle mass dramatically increases, but they take a hit in their speed. Goku demonstrated this to Gohan, during the Cell Game inside the Room of Spirit and Time. Vegeta also noticed this disadvantage during his training. Trunks however failed to spot this, and this led to his defeat by the weaker, but much slower Cell.
Achieved by: Goku, Vegeta, Mirai Trunks and Brolly
Super Saiya-Jin 2
This stage is very similar to SSJ, the only differences being lightning bolts that surround the Saiya-Jin and having slightly longer and stiffer hair. In this form power and speed are both increased over SSJ. This form was first achieved by Gohan whilst battling Cell, and is too triggered by extreme emotion.
Achieved by: Gohan, Goku, Vegeta, Brolly and Vegetto
This form is the strongest of all Saiya-Jin forms in the manga. Only Goku and the fused body of Gotenks can handle it. It is characterized by very long, full hair and the Saiya-Jin's eyebrows disappearing. Goku achieves this form whilst training in Heaven in between the Cell and Buu Sagas. This forms increases power and speed even more, but also uses a lot of power, so it cannot be maintained for very long.
Achieved by: Goku and Gotenks
Super Saiya-Jin 4
This is a form that causes some controversy, because as it doesn't appear in the manga some people say that it isn't a true form, but I am going to agree with the common consensus and say it is. In this form the hair stays blank and becomes long, but shorter than SSJ3. The Saiya-Jin's tail also grows back and red fur covers most of the upper body, the eyes turn yellow with red outlines and they have a hole in their palm form absorbing energy. Goku is the only true person to reach this form, but Bulma creates a device that allows Vegeta to go SSJ4 as well.
Achieved by: Goku, Vegeta and Gogeta.
This is the form that only Gohan reaches. He achieves it after his power up from Rou Dai-Kaioshin. During this state his power is fully maximized. So much so that he does not need to ever go Super Saiya-Jin to increase his power. It looks exactly the same as his normal state, as you can see from the picture. If the Rou Dai-Kaioshin had finshed the power up Gohan would have been stronger than SSJ3 Goku, but he never finishes because he is too lazy.
Achieved by: Gohan
When any Saiya-Jin with a tail is exposed to the light of a full moon, this sets off a chain reaction from a gland in their tails and causes them to transform into a 20-foot tall ape (Oozaru is Japanese for "Big Monkey"). This causes the Saiya-Jin's power to be multiplied by 10. With training this form can be controlled but Goku and Gohan were never trained this way so they "regained their Saiya-Jin savagery".
Achieved by: Any Saiya-Jin with a tail who saw a full moon
This is very similar to the regular Oozaru form except the fur is golden not brown. It is also of course much stronger. Once again this is a form that never appeared in the manga. To reach this stage a Saiya-Jin must have a tail and receive more waves than usual from the moon.
Achieved by: Goku and Bebi-Vegeta