King Vegeta is the father of Vegeta and Tarble and also the King of the Saiyans.
King Vegeta is the father of Vegeta and Tarble and is also the King of all Saiyans, Vegeta being older than Tarble is given the title of prince while Tarble is sent by his father to a distant star system for having no interest in fighting. King Vegeta was the commander of the Saiyans during their war with the Tuffles. King Vegeta wiped out the Tuffles and claimed the planet as Planet Vegeta. Before the destruction of Planet Vegeta, King Vegeta realized that Freeza no longer had any use for the Saiyans or that Freeza noticed a high power levels in low class Saiyan warriors which prompted him to destroy the Saiyan's planet, King Vegeta led a group of his top Saiyan warriors to fight Frieza in order to save Planet Vegeta. However, his plan failed and he was killed by a
single blow from Freeza. The witnessess of King Vegeta's death are Zarbon and Dodoria, apparently King Vegeta used to tell stories about The Legendary Super Saiyan to Vegeta and that one day Vegeta would reach that power level. King Vegeta also feared the power of Broly as of Broly's birth, his power level was equal to that of the infant Broly who possessed the power level of 10,000 which is regularly the power level of the strongest elite Saiyan warriors. King Vegeta fearing that when Broly becomes older he will use his monstrous power to overthrow him and orders that Broly be executed.
However Paragus, the father of Broly, pleads King Vegeta not to execute his son stating that he is only a baby and that he could be a great help to King Vegeta and the prince (Vegeta) however King Vegeta ignored Paragus' words and ordered that Paragus should be executed along with his son, their execution was followed by the destruction of Planet Vegeta.
It is never perfectly explained how King Vegeta became the king of the Saiyans, however it is possible that this is because he is the one who led the Saiyans into a successful war against the Tuffles and possibly because he was the strongest Saiyan at the time. King Vegeta showed little affection for Tarble as he
sent him to a remote planet for lacking the fighting ability and aggression needed to be a Saiyan warrior. Thus classifying him as a low-class warrior rather than an elite like the rest of his family. King Vegeta's wife is never mentioned or seen in the entire series and manga. The only full blooded Saiyan female to appear with her name revealed is Celipa, who is featured in the Bardock movie. However, there are Saiyan women who appear in the background during the movie. Big duh right there, how else would they re-produce.
King Vegeta possessed the power level of 10,000 which at the time was one of the highest power levels an elite could reach and the highest at the time, King Vegeta's power level was on par with that of Goku's father Bardock who also possessed the power level of 10,000. This is of course pure fan speculation.
King Vegeta was the strongest Saiyan at the time and his physical attributes were higher than that of every Saiyan, King Vegeta possessed superhuman strength, superhuman speed, superhuman reflexes, superhuman stamina, superhuman agility, energy manipulation, flight and being the King of Saiyans he had experience with the military and had great combat skill. A display of King Vegeta's power was when he easily disposed of Paragus with a simple energy blast with little effort however that is debatable as Paragus' power level is never mentioned and it is very possible that he is a weak Saiyan. Another great display of power in the anime is when King Vegeta destroys several planets and moons with a wave of his hand. However in the manga it is said that his son Vegeta surpassed him as a child.
King Vegeta looks exactly like Vegeta only he has has a an exceptional beard and his hair is brown and lighter than Vegeta's, King Vegeta's voice is also very similiar to Vegeta's being slightly deeper and they have the same accent. King Vegeta's armor has the Vegeta Royal Family Crest on the left side of his armor just as Vegeta did at the time. He also had a cape on his armor as Vegeta did as well.