For a Saiyan to become Super he/she must be pure of heart. They must be stronger then that of a normal Saiyan. And expereince extreme Rage to trigger the transformation (ex: when krillin dies by Frieza, Goku becomes full of anger and it triggers the Super Saiyan transformation). The reason Vegeta becomes Super Saiyan is because his heart is Pure.......PURE EVIL.
Super Saiyan Details: When a Saiyan goes Super his/her eyes turn Green, the body becomes bolder more muscular the hair turns Yellow and is pointed up (like Vegeta's).