While Goku has been a formidable fighter ever since his debut in Dragon Ball, the sequel series Dragon Ball Z revealed he was part of an extraterrestrial warrior race known as the Saiyans. Fierce, proud and capable of devastating attacks and combat transformations, the Saiyan royal family -- headed by King Vegeta and his son named after him -- were widely regarded as the most powerful of the Saiyans.
Here's how the royal bloodline gained its reputation as the most elite warriors in the entire Saiyan race -- and how Goku was able to surpass Prince Vegeta despite being regarded as a low-level commoner.
With Saiyan life centered around combat and conquest, it's natural that the strongest among the race would eventually be crowned as the civilization's rulers. King Vegeta was generally regarded as one of the most powerful warriors on the planet while Prince Vegeta was lauded as a prodigy, shown to be capable of incredible attacks and combat acumen even at an early age. However, the royal family was seen to employ more underhanded, murderous methods to ensure its supremacy remained unchallenged, with canonical and non-canonical instances involving the Legendary Super Saiyan Broly.
The non-canon 1993 anime film Dragon Ball Z: Broly - The Legendary Super Saiyan revealed that Saiyan scientists quickly noticed Broly's power level was an incredible 10,000, even as an infant. Fearing this growing power would eclipse him and his son, King Vegeta ordered Broly and his father Paragus killed, though both would survive the assassination attempt. The canonical introduction of Broly in 2018's Dragon Ball Super: Broly shows him and his father fleeing to the planet Vampa after King Vegeta notices the boy's immense power. This suggests the King previously employed similar tactics whenever he detected power levels that challenged his bloodline's reputation as the strongest of all Saiyans.
Goku was spared this fate because he was born with a relatively low power level, sent to conquer Earth since humans were identified as being relatively weak by comparison. However, he was depicted as a naturally gifted fighter after overcoming his mental Saiyan programming, trained by his adoptive grandfather Gohan before taking on Master Roshi and Kami as subsequent mentors. And while Goku was still relatively weak compared to more experienced Saiyans, he underwent experiences that eventually propelled him above Vegeta early in DBZ.
The Saiyan Saga kicked off Dragon Ball Z with the shocking twist of Goku sacrificing his life to defend Earth from his evil older brother Raditz, then arriving in the afterlife to train under King Kai. The godly training Goku received would significantly boost his potential but, perhaps more importantly, the Kaio-Ken technique learned from King Kai pushed Goku's body beyond its normal limits. This, combined with Goku's more innovative approach to training, would help keep him ahead of his longtime frenemy -- though Vegeta would close the power gap on several occasions.
While training Goku and Vegeta simultaneously near the beginning of Dragon Ball Super, the angelic attendant Whis perhaps said it best: Vegeta possesses a stronger tactical mind but overthinks his strategy, while Goku is more spontaneous and relies primarily on instinct. This distinction informs Goku's approach to training -- knowing how to get the best results to improve his body -- while Vegeta is often out for improvement through sheer power. And while both approaches have their respective merits, they ultimately let a 'commoner' like Goku catapult above the Saiyan Prince.