The Saiyans are one of the major races featured throughout Dragon Ball, with many of the franchises' most powerful figures coming from the alien species. But on top of all their impressive and amazing abilities, perhaps their most useful -- especially earlier in the franchise -- only comes into effect whenever Saiyans are almost killed.
What is Saiyan Power, the natural aspect of Saiyans that allows them to actually become stronger any time they come close to death?
The Saiyans from the Dragon Ball franchise are a remarkably powerful species, capable of reaching beyond the limits thought possible by most living beings. Saiyans are naturally strong and fast, and with proper training can hone those natural abilities to become unbelievably powerful.
With Ki attacks taught to even the children of the species, adults of the race are more than capable of destroying entire planets. Flight is a natural ability for them, as is enhanced strength, speed and durability over what regular humans are capable of -- along with enhanced metabolism and longer lifespan. They are typically born with a monkey-like tail, which allows them to achieve the powerful Great Ape form under the light of a full moon. They can even achieve various forms of Super Saiyan.
However, the most useful attribute of Saiyan biology is easily their ability to overcome their previous power levels through the simple act of healing from wounds. Referred to as the Saiyan Power, this element is a genetic trait that also carries over to half-Saiyans like Gohan, Trunks and Pan. This aspect of Saiyan biology allows them to recover from injuries at an abnormally fast rate, similar to a minor healing factor. But on top of that, the Saiyans also gain a power boost from the injury.
As their bodies heal, Saiyans develop natural resistances to whatever aspect of the attack damaged them, increasing the Saiyan's physical stamina and strength. The more powerful the attack that almost felled the Saiyan, the more dramatic the power boost will be after the Saiyan is finished healing. Although Saiyans can still suffer major injuries that require time to fully heal and can still certainly be killed, this ensures that any near-death encounter for a Saiyan has a long-term positive side-effect. This is one of the reasons that warriors like Goku and Vegeta have become so powerful: both heroes have suffered major injuries against deadly opponents, only to be healed and walk away with a massive power boost.
During the Frieza Saga, Vegeta even realized a way to utilize this aspect to his advantage: by having Krillin unleash a full-power attack and mortally wounding him, Vegeta was then able to get Dende to use his healing powers on him. This quickly restored Vegeta's health AND increased his power -- albeit not enough for him to be able to counter Frieza in his fourth form. However, there does appear to be a cap on how much a Saiyan can use this ability. During the events of the Buu Saga and Dragon Ball Super, Saiyans are restored to full health but receive no such power upgrade. It's suggested that this may be because the core heroes have reached such a natural point of power that any boost from just their Saiyan biology is minimal compared to their current potential.
Still, the Saiyan Power quickly helped ensure the heroes stood a chance against the powerful enemies they encountered throughout Dragon Ball Z. Without it, it's unlikely Goku and his allies would have ever lasted long enough for him and the other remaining Saiyans to ever get powerful enough to reach Super Saiyan.
