Is a new transformation similiar to the Super Saiyan form
Super Saiyan 2 (Sūpā Saiya-jin Tsū) (abbreviated as SS2 or SSJ2) is an entirely new transformation, very similar to Super Saiyan in appearance and even attainment. In order to achieve this transformation, the Saiyan must have experienced an intense emotional upheaval, much like the original Super Saiyan transformation, but to a greater extent. Because of the intense emotion required to initiate the transformation, any mastery of the Full-Power Super Saiyan state, and mental stability attained therein, is negated, and the naturally remorseless nature of the Saiyan race is magnified, requiring them to remaster the personality changes. Even the gentle, polite Gohan becomes a hotheaded, merciless fighter after transforming, opting to torture Cell rather than kill him despite his father's pleas.
The golden hair brought on by the original transformation becomes longer and rigid; any hair that hung loose in the Super Saiyan transformation (in the case of Goku or Gohan, because Vegeta lacks any strands in front of his face, as does Gotenks) is now raised. Increased energy radiation causes the aura to take on a jagged, fierce flame-like appearance rather than smooth or flowing. Because the energy output and radiation is higher than that of any Super Saiyan, the aura pulses at a higher frequency, like the type 2 aura, at least in the manga. Muscle mass is barely increased, but energy output is multiplied manyfold; such that greater amounts of the Saiyan's increased energy capacity can be used at once for much more powerful energy attacks, and even speed is augmented incredibly by the boost, unlike the second and third grade states which cause the muscle mass to swell and thus impede the user. Another notable feature of the form featured in almost all Dragon Ball Z games and some scenes of the show as well as pictures from the Manga, is the electrical aura, many electrical sparks (normally blue, gold in Dragon Ball Z Budokai 3) that constantly surround the body — especially when charging up. However, they aren't always present (especially in the anime; Gohan and Vegeta aren't the only ones who lack the trait completely in some episodes), and some fans sometimes mistake a charging Super Saiyan for a Super Saiyan 2 due to the lightning sparks, such as Goku Jr. in the A Hero's Legacy special and Trunks and Goten in Dragon Ball GT. However, the one and only time the lightning aura is depicted on a first level Super Saiyan is with Vegito; in the manga, they never reappear in his aura after his transformation, nor are they ever shown on any other first-level Super Saiyan.
This form was chronologically achieved by Gohan after witnessing the Cell Jr's attack on the Z Fighters and Android 16's death at the hands of Perfect Cell. The power of the Super Saiyan 2 skyrockets far beyond the power of Super Saiyan, Super Saiyan 2nd and 3rd Stage, and even Full-Power Super Saiyan. Perfect Cell and Bojack, who could both easily dominate those of the Super Saiyan and 2nd and 3rd grade levels and were somewhat stronger than Full-Powered Super Saiyan even without accessing their full strength, were easily slaughtered by Gohan in his Super Saiyan 2 form. In addition, when Cell produced seven Cell Jr.s, each of the Z Fighters were unable to fight evenly with even one of them (including Goku, Vegeta and Future Trunks, but the former was exhausted from his previous fight with Cell himself), though Gohan killed all of them effortlessly on his own in simple, single blows, and was unaffected by even their most powerful attacks.
Gohan proceeded to fight and severely injure Perfect Cell with just two hits, even when the android unleashed his full power. However, in a Kamehameha struggle against Super Perfect Cell, Gohan had a very hard time due to having his left arm broken by Cell's attack when he defended Vegeta after he charged to try to avenge his son (a blast which was stated to reduce Gohan's power output to less than half its normal amount). It was only a little help from Goku and Vegeta that Gohan needed to summon enough power to destroy Super Perfect Cell. A few months to a year later, Gohan used his Super Saiyan 2 power on Bojack and defeated the tyrant and his henchmen in a matter of seconds.
Seven years later, Gohan demonstrated the Super Saiyan 2 transformation to Kibito in the World Martial Arts Tournament, but this is the last time he uses this form. It is also suggested that he used this form to battle Broly when he reappeared in Broly: Second Coming. However, he was outstripped by the Legendary Super Saiyan's immense power, although this was partially because of Gohan's lack of training up to that point that made him noticeably weaker, and since Broly survived being punched in the gut by Goku after he was powered up greatly, Broly's power most likely increased substantially after recovering from near-death (as all Saiyans do). Though, to Gohan's credit, his blows managed to cause Broly a noticeable level of injury (whereas nobody in Broly's first appearance could damage him in the least), as Broly was knocked back when he was struck by Super Saiyan 2 Gohan. The Daizenshuu states that Gohan also went Super Saiyan 2 against Dabura. Vegeta mentioned that Gohan's Super Saiyan 2 form was now much weaker than when he fought Cell due to his lack of training.
Following the events of Majin Buu, seven years after the Cell Games, Goku and Majin Vegeta fought each other in the Super Saiyan 2 form. The arrival of Majin Buu is shown to far surpass the power of a Super Saiyan 2; Vegeta did little damage to the monster, even when he expelled all of his ki at once in a single, devastating, suicidal explosion. During the fight with Kid Buu, Goku and Vegeta battled as Super Saiyan 2 for a short time (in the anime only, because Goku goes straight to Super Saiyan 3 in the manga, stating that he'd have no chance if he doesn't use his full strength), although Goku ended up transforming to the Super Saiyan 3 level.
It is stated in Daizenshuu #7 that Gotenks can theoretically transform into Super Saiyan 2. However, due to bypassing the Super Saiyan 2 form when transforming from a Super Saiyan into Super Saiyan 3, his complete Super Saiyan 2 form is not ever fully revealed, and remains unused.
Furthermore, during Dragon Ball GT, Goku, Vegeta, and Gohan all used the form from time to time during battle. However, by this time it had been majorly outclassed by Super Saiyan 3 and Super Saiyan 4. Most assume Goku's first usage of the form to be when he fights Rildo's final form on the metal planet due to the intense aura he manifests as well as the forehead strands disappearing the way that his second-level hairstyle depicts, as shown above; however, he only stays transformed for a very brief time by simply powering up to the state to vaporize a metal ball that he shot. Goku also seemed to access the state momentarily again, as he was about to transform into Super Saiyan 3 to fight a futile fight against Super Baby Vegeta, as shown from the lightning sparks, aura, and the glow effect primarily present with higher-level Super Saiyans (and thus in the creases of the body, clothes, and hair), but due to him not assuming his level two hairstyle on any other occasion, it remains dubious as to if he used the state any other time.
One strong case for Gohan and Vegeta's usage of the state is when all four Saiyans except Goku power up with Majuub to fight Super Android 17, where the robust aura associated with the state appears on Gohan and Vegeta, but not on Goten or Trunks, both of whom never assumed the state outside of their fusion; the sparks however are not present, and Goku only appears and fights in the first level state before being forced to use Super Saiyan 4 without any usage of either the second or third levels.