Vegeta is the prince of all saiyan's. About a third to half way through DBZ he becomes able to turn into a Super Saiyan along with Goku, Trunks and later Gohan. He was one of the four surviving Saiyan's after Frieza destroys the Vegeta's home planet.
Saiyan's were Frieza's henchmen for a while until Vegeta's dad King Vegeta tried to overthrow Frieza, in anger Frieza destroyed planet Vegeta. As a child Vegeta's was taken by Frieza so was off doing a mission for Frieza when his father and the planet were destroyed. Vegeta always new that Frieza destroyed his home planet though so always thought about over throwing Frieza one day.
Vegeta was always thought to be the most powerful saiyan left and prided himself on it, that changed when he met Goku and has always been trying to do better than him ever since that day.
At the beginning Vegeta was the Z team's enemy that changed when they had the common enemy of Frieza. He was thought at the beginning to be the bad member of the z team and doesn't have any morals but he occasionally shows his emotions and it is realised that he does love his friends and does actually like Goku a bit, although he tries not to show this and does not say it either.
He marries Bulma and they have two children Trunks and Bra. The word Vegeta is a play word on Vegetable and before the Saiyan's took over planet Vegeta it was actualy called planet Plant they renamed it.