Zenkai (全快; lit. "Full Recovery") is an ability that is genetically exclusive to Saiyans. The ability is a genetic trait that allows a Saiyan's power to increase substantially after recovering from near fatal injuries. This trait is not exclusive to pure-blooded Saiyans like the lack of hair growth is, as even clones with partial Saiyan genetic data have this ability as well. Examples are half-breed Saiyans, such as Gohan, and the biological menace Cell.
During the Namek and Frieza Sagas, Zenkai is an extremely important element of the series, which allows Goku and Vegeta's powers to increase in order to "catch up" to Frieza and his men. Zenkais seem to stop occurring after the Frieza Saga (when most of the Saiyans have already become Super Saiyans), but returns for one single instance in the Cell arc when Cell returns from his botched self-destruction on King Kai's planet, his power amplified to the point that he has high enough amounts of energy surrounding his body to mimic the style of aura that a Super Saiyan 2 has.
When Vegeta recovers from his first fight with Goku, his power level went from 18,000 to 24,000, and another was received after being nearly killed by Zarbon's transformed state and healed in a Medical Machine, which the Daizenshuu estimates to have augumented his strength to roughly 30,000. After Vegeta was again near-fatally injured after fighting Recoome, his strength strangely skyrockets enough to have an even power struggle with Frieza's first form (stated to be 530,000), and another, final Zenkai increased his power enough to see Frieza's movements and Death Beams at first, before he begins to put in even more effort. While the dub incorrectly implies that Zenkais only occur with Elite Saiyans, Goku, whom Vegeta and Nappa repeatedly refer to as low-level trash ("clown" in the dubbed anime), receives them many times. This is confirmed by how Krillin states that it was weird how Goku always seemed to grow ever stronger after recovering from fighting devastating battles against powerful foes, how Goku is able to learn and tap into the power of Zenkai's during his gravity training (which in total boosted his power from "over 8,000" to 90,000 in the Namek saga, as 180,000/2 = 90,000), and how Goku's powers again receive an enormous boost after recovering from his battle with Captain Ginyu in the rejuvenation chamber. To force a boost in power during the battle with Frieza, Vegeta tells Krillin to brutally injure him and have Dende heal him, which after greatly increased his power.
Goku experiences a Zenkai upon eating a Senzu Bean in the movie Dragon Ball Z: Cooler's Revenge. In Dragon Ball Z: Broly - The Legendary Super Saiyan, it is stated that Broly survived his execution and grew stronger due to his Saiyan genes. In Dragon Ball Z: Broly - Second Coming, it is implied that Broly grew significantly stronger while in a coma for seven years after being beaten by Goku and recovering.
Bardock (Goku's Father) gets a massive Zenkai boost when he is killed by Frieza and goes back in time (No clue how) to kill Chilled (Frieza's great-something ansestor) The Zenkai boost and need to be stronger made Bardock go into Super Saiyan ( SSJ1)