To transform into a Super Saiya-Jin, the Saiyan must be able to build up the anger inside. Let the anger to increase the desire of revenge therefore pushing the Saiya-Jin's power level suprassing the maximum limit. If the process is successful, the Saiya-Jin should have gold hair which sticks up and eyes turning green. They also have this glow of golden light wraps around the Saiya-Jin. As a Super Saiya-Jin, it increases the power, speed and agility. This was how Gokou killed Freeza in the DragonBall Z series (Freeza Saga).
Advantage:

It increase the overall power.
Disadantage:
None
Interesting Facts:
Gokou was the first Earth-raised Saiya-Jin to become a Super Saiya-Jin, and maybe the first Saiya-Jin other than Brolly (The ancient legenday Saiya-Jin). Vegeta got jealous seeing Gokou as a Super Saiya-Jin, therefore he quickly went back to training which made him the second Saiya-Jin to turn into a Super Saiya-Jin.
Goten is the first ever Saiya-Jin who has the natural ability to turn into a Super Saiya-Jin, without pushing himself to the limit. Gohan found out this ability while he was training with Goten, while Gohan was explaining to Goten that Goten couldn't possible become a SSJ without pushing himself, then all of a sudden Goten just turned into a SSJ. Gohan was very suprised.
Gohan turned into a Super Saiyan by training with his father Gokou while preparing the for the Cell Game during the Cell Saga.
Gohan's death by Androides 18 and 17 triggered young Trunks' anger and the desire for revenge, hence transformed Trunks into a Super Saiya-Jin. [note: see Story of Trunks]
Believe it or not, Pan (Gohan and Videl's daughter) and Bra (Vegeta and Bulma's daughter) also have a good chance of becoming a Super Saiya-Jin. However, one disadvantage is that they are females, therefore it makes them harder to reach to the level like Gokou, Gohan, Goten and the rest of the male Saiya-Jins. I guess we never know, maybe one day, one of them might become the first even female SSJ...
Vegeta was the first to reach SSJ Level 3, by successfully adjusting his strength/speed/agility. He trained and experimented with his body and worked out most optimal strength/speed/agility for himself, and gained a huge boost in each of the three catagories. The results was clear, he had no trouble beating up incomplete Cell and sent him packing home to get more training. This transformation by Vegeta, by honoring his own success, he named this transformation as "Super Vegeta".
A theory that, the transformation from a Saiya-Jin to a Super Saiya-Jin, their power level is increased by a factor of 50.
Interesting thing is that, SSJ is also known to have 3 different levels of its own. SSJ level 1, SSJ level 2 and SSJ level 3. SSJ level one is the most complete SSJ transformation, SSJ level 2 is increased speed and strength, and SSJ level 3 is increased strength, but decreased speed.