"Super Saiyan" is a legend passed down among the Saiyan, of a genius fighter among the Saiyan, who comes along only once in every thousand years.  It is said that the Super Saiyan is cruel and powerful; so powerful, in fact, that there can be no one stronger in the universe.

The Saiyan's hair turns gold, and the eyes turn bigger and green.  The higher the Super Saiyan's power level is, the longer the hair grows.  The highest Super Saiyan power level was 3; which Goku reached.  There is a "Super Saiyan 4" in GT, but it is not a true "Super Saiyan" form because the hair remains black.  It looks more like a half monkey, half Saiyan.  Hair grows on the chest, a tail grows, eyes turn evil like.
 

Qualifications of becoming a Super Saiyan

1.  Must be at least partly Saiyan. 2.  Must possess at least enough power to potentially surpass the      limits of a common Saiyan. 3.  Must be pure/have a pure heart. 4.  Extreme rage triggers the realization of being a Super Saiyan. 5.  After the first transformation into a Super Saiyan, the trans-      formation becomes relatively easier each additional time.

How did Vegeta turn Super Saiyan?

Referring back to "Qualifications of becoming a Super Saiyan", Vegeta satisfies both the first and
second qualifications easily.  As for the third and fourth rule:

Vegeta: "I  am  pure EVIL!!"
(or   "I  do have a pure heart...  It's just pure evil!!")