Super Saiyan
A Super Saiyan is a Saiyan that has gone beyond their limits through training. The Super Saiyan transformations yields a 50x increase in power with each level. Goku is the first to achieve this level. Any saiyan with enough power can transform with sufficient emotional need (usually the pain of loss).

There are 4 different super saiyan levels:

1. Super saiyan

1a., Ascended Super saiyan, achieved by sacrificing a bit a speed for a large power increase, 1b; Ultra Super Saiyan, sacrificing more speed for an even bigger increase in power. Both states requires going beyond normal super saiyan limits.

2. Super Saiyan 2: This state is achieved when a saiyan goes beyond his natural super saiyan limits, and the user is enraged enough to achieve this transformation; usually through the pain of loss.

3. Super Saiyan 3: This state is achieved when a super saiyan 2 has surpassed it's natural limits and a deeper power buried far within is available. Takes a lot of practice to transform quickly. It requires an immense amount of power to sustain, and constantly drains the users energy.

4. Super Saiyan 4: The pinnacle of Saiyan power, this state is achieved when the user is transformed into a Golden Great Ape (A super saiyan who transforms at the full moon) and becomes conscious of his new power. As with all super saiyan transformation, the user's power is increased 50 fold.
Goku and Vegeta are the only full-blooded Saiyans that can transform into Super Saiyans in canon series. Broly is the other, but only in the films.