Vegeta is the Price of Saiyans. He is King Vegeta's son. Vegeta was born on the Planet Vegeta. Vegeta is an Elite Saiyan Warrior, because he was born very strong. Due to his strength Frieza wanted him as his assistant. Frieza had threatened to detroy the planet, therefor King Vegeta swapped Vegeta for the safety of his planet. After Frieza had Vegeta, he destroyed the planet anyway. Vegeta, together with Nappa, Raditz and Goku are the last remaining saiyans after the planet was detroyed. Vegeta had always been a loyal submissive to Frieza, because he thought that his planet was destroyed by a comet. When Vegeta found out that Frieza had destroyed his planet, Vegeta got angry and wanted revenge on Frieza. Vegeta wasn't strong enough to defeat Frieza. But his desire for revenge remained in him. When Vegeta came to Earth to search for the Dragon Balls he met Goku. Vegeta was defeated by Goku, but Vegeta knew that there was something special about Goku. On Namek Vegeta was killed by Frieza, when Vegeta told Goku that Frieza had killed Goku's father and Vegeta's father. Goku wanted revenge for his father, all saiyans and the rest of the universe.
Goku became a SSj due to that. Vegeta was angry because Goku had become a SSJ before he did and started training intensifly. Vegeta wanted to become stronger than Goku. Vegeta became a Super Saiyan too and he did become stronger. But that didn't last very long.
Vegeta returns to be a bad guy in the Buu saga, because he discovered that there was a lot of evil in Vegeta. Vegeta becomes a SSJ2 in his Majin stage. Vegeta turns back into a good guy again in the Buu saga too. Because he wants revenge on Goku he murges with Goku as both of them aren't strong enough to defeat Buu. Vegeta then becomes Vegetto. This happens because of a pair of earrings. In the Buu saga Vegeta is so mad that he blows up himself trying to defeat Buu. Vegeta doesn't becoma a SSJ3. But in Dragon ball GT he does become a SSJ4. That happens because Bulma gives him something called Brute Ray. In the GT series Vegeta is about 41 years old. He became a little bit of a gentler man and lost most of his big ego that he had in Dragon Ball Z. But he is still cool. Vegeta becomes Bebi Vegeta in the Bebi saga from the GT series. He transforms and wants to destroy the saiyan.