Vegeta is almost a polar opposite of Goku, and is one of the most changed characters in the entire series, which makes him a perfect candidate for Goku's friend/rival.
Vegeta is a Saiyan, and hails from Planet Vegeta, just like Goku. He was born in Age 732 and his father is King Vegeta, who was the king of the Saiyans. He was trained throughout his life and after proving himself he went to take over a planet for Frieza. However, while he was off of Planet Vegeta, Frieza destroyed it, leaving few survivors. Afterwards, Frieza keeps him along with Raditz and Nappa 
alive to do his bidding as soldiers in his army. In the manga, Vegeta is angry to serve Frieza, and us reluctant to take orders from his right hand man Zarborn. He has stated Zarbon mocked him and worked him like a slave. Even though Vegeta was Frieza's underling, he still proudly calls himself "the prince of all Saiyans." Despite his father's death, he does not call himself the "king of all Saiyans" this could be out of respect for his father, refusal to admit the genocide of the Saiyans, or simply because he was never formally declared king. Vegeta was said to have tried to assassinate Frieza several times, but stopped as he realized how much power Frieza had when witnessing him destroying a planet. He was sent on many missions, one of which was alongside Raditz and Nappa, and Vegeta was now growing into an adult. They were sent to Planet Shikk, but when they came back Vegeta had to hold both himself and Nappa back from attacking Zarbon Dedoria and Frieza for mocking them about the way they took over the planet. Vegeta was shown to have verbal fights with the Ginyu Force member Guldo about who is more important in Frieza's eyes. Vegeta grew up on Frieza's ship and never left until he came to Earth and attacked Goku and his friends, then he became a renegade of sorts as he abandoned Frieza completely until he was defeated.
Throughout the series, Vegeta is arrogant about himself, and bitter about the disgraces the Saiyan race has gone through under Frieza. He has a large sense of superiority and holds himself above almost anyone he fights. He has immense pride in his Saiyan heritage and believes the Saiyans to be the most powerful race in the universe. Anyone who disgraces the Saiyans or pretends to be a Saiyan (such as Zamasu did as Goku Black) he will be infuriated. At first, Vegeta was introduced as a villain, doing evil actions and anything heroic he did was only done because of self-interest or revenge, not because it was heroic. Vegeta is a tactical fighter, who will often rush into a fight with no hesitation, but because of this and his arrogance and rage, he will underestimate his opponents, which will often lead to his defeat. Vegeta is smart and serious, rarely spouting humor or laughing unless he finds mocking someone else amusing. When he uses humor outside of mocking his enemies, it is usually very dry and blunt, and typically darker based humor. These egotistiacal and arrogant traits will often overshadow his positive 
traits, and even at first these traits did not seem to exist at all. Vegeta stayed cold and aggressive for the first parts of the series, but he began to change in the Namek Saga. He spared Gohan and Krillin even though he had promised to kill them after he was defeated on Earth, and formed an alliance against Frieza, although this was slightly out of desperation. Throughout the Frieza Arc he was shown to kill his enemies without mercy, but when he attacked Gohan while looking for the Dragon Balls, he only kneed him in the stomach instead of completely killing him, showing his first ever sign of mercy to someone. Another sign of mercy was letting Captain Ginyu live after he had used his body change ability to accidentally swap bodies with a Namekian frog. This wasn't complete mercy however, as he thought being swapped with a frog was punishment enough and he didn't want to get frog guts all over his boots. However, his biggest change in personality came when he was defeated at the hands of Frieza. He began to cry as he explained to Goku that he was forced to be ruthless, and he never had the chance to be anything else. He begged Goku to defeat Frieza, and bring honor to the Saiyan race. From then on after being revived by the Dragon Balls and brought back to Earth, he began to slowly soften, but still kept his arrogance and pride.His ruthlessness was all but gone now, and he no longer wished to kill Goku's friends, but he still had a grudge with Goku. He now had a family, Bulma became his wife and they soon had a son, Trunks. He was shown to care about his family, as there have been several times in the series where he has become angered due to Bulma being taken advantage of or hurt. He even begins to train his son when he grows up enough, and he cares very much for his later daughter Bulla (or Bra in the original Japanese version). Vegeta even admits to how much he cares about his family after he gains his senses despite being controlled by Babidi. He reveals that he let himself be controlled in order to restore his former evil behavior, which he believed would reveal his true power. Goku thought he was bluffing, but he showed the truth in the statement by hugging Trunks for the first time and saying that he was proud to be his father before sacrificing himself in a final attempt to destroy Majin Buu.
Vegeta may have changed a lot since his first appearance in Z, but there are several traits that have stayed the same. Vegeta is still as stubborn and headstrong as ever, and always tries to hide the softness he has gained from others. He acts cold to most people, but to Bulma, Trunks, and occasionally the other Z-fighters, his developed traits will show through. His pride is still the same, and while it has 
been challenged, he's never lost it. He still proudly calls himself the prince, and never lets anyone get away with insulting the Saiyan race. And the most prominent thing that has stayed the same throughout Vegeta's development is his rivalry with Goku and his determination to surpass him. Goku may not see it as a rivalry, but Vegeta does. He will constantly train and push himself with only the goal of surpassing Goku in power. He has ignored several things in his life for training, almost including his own wife's birthday celebration. He always gets angered when Goku shows that he's stronger than him, like when he went Super Saiyan. In fact, the only reason Vegeta was even able to go Super Saiyan in the first place was because of his rivalry with Goku. These traits are still present in Vegeta and haven't been and probably never will be changed.