Super Saiya-jin (SSJ) is a special technique only available to Saiya-jins and hybrids of Saiya-jins (Gohan, Trunks, Goten). It was once considered a myth by the Saiya-jin race from an incident that happened 1000's of years ago. The first SSJ to actually be seen was Goku, with the help of extreme anger (from the death of Krillin) Goku managed to pull off the transformation against Frieza. The SSJ transformation and all it's improved forms involve a great increase in power, speed and agility. The only exception to this Super Saiya-jin dai 3 dankai, which involves a great increase in power but actually decreased the speed, giving the user (Mirai Trunks) a great disadvantage against against his opponent.

Below is a description of each and every form of Super Saiya-jin that exists in db/z/gt. Click on the name of the form to get an example of the form.

Saiyan - Saiya-jin.
This is your average Saiyan, they used to be classified into groups based on their power level at birth, for example; if you were born with a pl [power level] of 1-5 [like Goku] you were classed as a 3rd class saiyan. If you were born with a pl of over 100 you were a saiyan elite [like Vegeta] The most powerful technique a saiyan has is called Zankai, it basically increases their power greatly every time the are injured or a close to death, this has allowed low class saiya-jins to become 10 times stronger in just a few days making the Saiya-jins a great threat to any enemy.

Oozaru.
The main features of this form is the fact that they become a 40ft tall giant ape, this form increases the Saiyans power by exactly 10 times. For a saiyan to go Oozaru they have to have their tail and they have to be exposed to 17 million waves that can only come from a full moon, no matter the size of the moon.

False Super Saiyan - Giji Super Saiya-jin.
False SSJ is exactly what it sounds like, it is "the false super Saiyan". This form was only ever attained by Goku in movie 4, he had the golden aura as usual, but no golden hair, think of this as half-way to being a super saiyan.

Super Saiyan [SSJ] - Super Saiya-jin.
For a Saiyan to become a SSJ they needs to become angry to start the reaction. For example, when Goku went SSJ he was angry at the death of his friend Krillin. When future Trunks first went SSJ he was angry at the death of his life-long friend and mentor Gohan, at the hands of the androids. When they go SSJ he/she experiences a great increase in power, a large golden aura surrounds them, their muscle mass is increased and their hair goes a bright gold and stands on end.

Ascended Super Saiyan [USSJ1] - Super Saiya-jin Dai 2 Dankai.
This is one stage beyond SSJ, the user gains a big increase in power but they suffer one big side effect, they lose some speed. The first to achieve this form was Vegeta after he emerged out of the Room of Spirit and time [ROSAT]. Ultra Super Saiyan [USSJ2] - Super Saiya-jin Dai 3 Dankai.
This is one step beyond USSJ1, it has even more power, but even less speed, it was first used by Mirai Trunks against Perfect Cell, despite him being stronger he still lost because he was too slow. this form involves a Big increase in the users muscle mass and their hair spikes up in random directions.

Full Power Super Saiyan [FSSJ]
This is not a level of Super Saiyan; it's a way of making SSJ more powerful. Usually to become a Super Saiyan, a Saiyan must use a large amount of his ki (about 35 to 40%) Keep in mind that the transformation dose not give the Saiyan more ki, it just makes whatever ki he has left more powerful. A full power Super Saiyan has learned how to stay transformed all the time and dose not have to use any of his ki to power up. Therefore, an full power Super Saiyan could be up to twice as powerful as a normal Super Saiyan, with the same base power level.

Super Saiyan 2 [SSJ2] - Super Saiya-jin 2.
SSJ2 is obviously the stage after SSJ1. It was first achieved by Gohan against Cell, Cell cruelly killed android #16 during his speech to Gohan. This enraged Gohan so much sending him to a new level of SSJ. When they become SSJ2, they receive yet another huge power gain, their hair has more visible depth, electricity is visible around their body and they gain even more muscle mass.

Super Saiyan 3 [SSJ3] - Super Saiya-jin 3.
In my opinion, this is the coolest of the SSJ levels. When you go SSJ3 you grow long gold hair [as usual] but it grows down their back, they also lose their eye brows. As usual they get another great boost in their power. The first to go SSJ3 was Goku during the Buu saga, the biggest downside to SSJ3 is the enormous drain in energy they receive. After going SSJ3 for five minutes Goku was completely knackered, he had trouble staying on his feet.

Super Saiyan 4 [SSJ4] - Super Saiya-jin 4.
This transformation is not even considered real by some. This is because all of the Dragonball/Z series was drawn by a man called Akira Tokiyama, he was also the creator and writer of Dragonball/Z. But he did not do anything for DBGT which was the only series where SSJ4 was featured. SSJ4 is definitely the weirdest looking of all the SSJ levels. When they go SSJ4 their hair remains black, no golden aura is created, and they grow red fur all over their body except their chest. To go SSJ4, first they obviously need sufficient power, they need a saiyan tail or the reaction will not happen. Remember it is not considered real by some!