A more powerful form of a Saiyan that un-limits more power.
For hundreds of years saiyans were a warrior race in the universe. Their power was focused on conquering planets for profits and for personal gain. Saiyans naturally could transform into Giant Apes thus helping conquer unsuspecting victims but the idea of a super saiyan was nothing but legend and rumor. This legend permeated throughout the universe but still was never seen and rumored to only occur every few thousand years or so.
In order to transform a saiyan must be at a high level of power and use their anger and rage to turn. Goku was in the heat of battle with Frieza, at the time the most powerful being in the universe seeking immortality, and needed to overcome his normal saiyan form to save his friends and the universe. This need caused the transformation, boosting Goku's power very high as well as making his stronger and faster. The most distinct feature of the super saiyan is the change in appearance, usually a yellow glow and blonde hair.
Goku was the first saiyan seen to transform but he wouldn't be the last. After some more training Goku would return home to Earth and be greeted by future Trunks, Vegeta's son from the future sent back to warn Goku and his friends. Seeing another super saiyan would goad Vegeta's ego and pride pushing him to train until he too was a super saiyan. Gohan would soon be next and soon becoming super saiyan was inherent in Goku and Vegeta's families, something learned at a very young age and achieved almost easily.
A level right before super saiyan when the saiyan boosts their power to incredible levels but with out further stimuli of anger and desperate need the saiyan only barely resembles an actual super saiyan. The assumption that Goku achieved this using his kaio-ken technique (a power boosting technique) most likely occurred because of the lack of information of what a true super saiyan is before Goku finally transformed for the first time.
Another level of false super saiyan but this time something achieved once the saiyan has become a super saiyan. While waiting for the Cell Games to commence Vegeta, Trunks, Goku, and Gohan all spent their due time training. Vegeta and Trunks trained by themselves and achieved the ascended super saiyan status. This state made them very powerful but heavily increases their muscle mass which slows them down to a detriment. This form is usually achieved from trying to boost one's power too much in hopes of turning into super saiyan 2 and was even achieved by Goku while training with Gohan right before the Cell Games.
The final form of the first super saiyan level which mainly maintains the blonde form at a comfortable level for the user. This stage sustains the power, strength, and agility of the super saiyan level but doesn't require the power level spike nor the anger that the original transformation required.
The next real transformation in the super saiyan level which gives the saiyan another huge boost to power, strength, and speed similar to the boost from normal to super saiyan does. Gohan was the first to achieve this level while fighting Cell in the Cell Games. While training with Goku beforehand, Goku sensed Gohan's inner power buried inside and was confident with his friends and family in danger that power would come out. Mostly right Gohan transforms when he witnesses Android 16 die at Cell's hands. The change looks very similar to the super saiyan form but with small electrical bursts and turns the personality into more ruthless and aggressive.