Super Saiyan (超サイヤ人, Sūpā Saiya-jin) is a powerful and legendary transformation that the Saiyan race can obtain. This transformation is obtained by extremely powerful Saiyans or Saiyan hybrids.
According to Vegeta, there was a legend among Saiyans that once every 1,000 years, a Super Saiyan, whose power is beyond imagination, will appear. Freeza, a Frost Demon emperor, in many universes, became paranoid over the Super Saiyan legend, which caused him to eradicate the Saiyan race, leaving only six Saiyans alive (Son Gokū, Vegeta, Nappa, Raditz, Paragus, and Broly). Twenty-five years later, in a majority of universes, Son Gokū put an end to Freeza's tyranny after obtaining the Super Saiyan transformation due to his best friend, Krillin, being killed by the tyrant.
When a Saiyan transforms into a Super Saiyan, their hair turns golden blonde-like, and their becomes much spikier. Their irises become teal, and a golden aura grows around them.
List of Saiyans who obtained Super Saiyan
Gokū, Vegeta, Gohan, Goten, and Trunks from Universe 18, due to the Universe being apart of the original story line. Same for the Gohan, Goten, and Trunks apart of Universe 16, along with the Trunks from Universe 12 and the Broly from Universe 20.
Vegetto of Universe 16, who obtained his from being a fusion of Gokū and Vegeta, as both of them originally obtained Super Saiyan. Gotenks (from Universes 16 and 18) also obtained this form from being a fusion of Goten and Trunks. Bra and Pan from Universe 16 gained there's easily from their father's enormous power, plus constant training for them to unlock their super forms.
Bra of Universe 16 obtained the transformation at merely six-years-old, from experiencing intense emotions and insecurity from her thinking she's weak.
Vegeta and Kakarotto from Universe 13 unlocked the form as well. Vegeta discovered it and beat Freeza and sometime after defeating Freeza, Kakarotto achieved it. It's unknown how exactly how they obtained the form, other than possibly using the Namekian Dragon Ball Balls.
Baddack from Universe 3 also unlocked this form. It is unknown how, but speculated that he gained it through witnessing his race being destroyed by Dr. Raichi. Ghost Broly and Ghost Vegeta of Universe 3 can also use the form.
Pan also unlocked the form after Kakarotto brutally beats her. While in the manga, Kakarotto accepts his "defeat", in the novel, he goes after her for revenge. The minicomic, goes even further by showing her as a SSJ3, saying that she wasn't annoying (referring to her GT counterpart).
The character designs also show that our Bra can also go SSJ, despite hating this form (due to beating "mindless"), and Salagir/Gogeta jr stating that she might never reach that form despite being drawn
Methods of obtaining Super Saiyan
For full-blooded Saiyans, the transformation is usually unlocked by experiencing extreme emotions or stress. However for most half-breeds, they seem to unlock the transformation a lot simpler than their Saiyan parent. Goten, for example, obtained the transformation at only seven-years-old, with little-to-no effort while training with Chi-Chi. Bra of Universe 16 obtained the form at six-years-old, becoming the youngest Saiyan to obtain the transformation. She obtained it, like Gohan, through heavy emotions and stress.
Interestingly, Vegeta has achieved this transformation the most out of all the Saiyans in all the universes, achieving the state in universes 3, 4, 6, 7, 11, 12, 13, 14, 15, 16, 17, 18 and 20 (thus he doesn't have it in universes 1, 2, 5, 8, 9, 10, and 19). This is quite impressive since Vegeta had considerable difficulty achieving it in Universe 18.
Other forms of Super Saiyan
3rd Grade Super Saiyan
Primarily referred to by fans as "Ultra Super Saiyan". Trunks from Unverse #12 has used this form to demonstrate his Full Power to Number 16. Since the main storyline is still here, Universe 4/16/18/20 in which the Cell Saga is all un-effected by respective events but this has not been used in a fight. Trunks managed to master the transformation by quickly activating it from his Super Saiyan form whenever he needs to attack, compensating from the loss of speed and rapid power output.
Legendary Super Saiyan
Only Broly can utilize this form. Legendary Super Saiyan has been altered from previous DBZ movies in that the form allows the user to be completely invulnerable to attacks and also spontaneously rise in battle power while battling powerful opponents. The only way to get them to revert to Normal is to cause intense psychological shock (Such as the Sun/A comet/star).
Access to the legendary Super Saiyan form also grants access o the normal Super Saiyan form.
When the user becomes mad with rage, they are incapable of rational thought or calming down. The user's life span is dramatically reduced. Broly is only alive because his power was growing in suspended animation. As an extension of the user's life span drastically being reduced, the Saiyan's own power destroys him ultimately in time, because the body cannot keep up with the power increase. Apparently subsequent generations have a better tolerance for this weakness, as each Legendary Saiyan lives longer than his predecessor.