From Dragon Ball Encyclopedia, the ''Dragon Ball'' wiki
Zenkai (全回, Zenkai) is a supportive technique 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 as 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 Son Gohan and the biological menace Cell.
For example, when Vegeta recovers from his fight with Goku, his battle power went from 18,000 to 24,000, and another was received after being nearly killed by Zarbon's transformed state and healed, 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 Freeza's movements and Death Beams at first, before he begins to put in even more effort. While the dub incorrectly infers 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 Kuririn 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 Zenkais during his gravity training in the Gravity Chamber (which in total boosted his power from "over 8,000", 90,000 during 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 Ginyu in the Medical Machine. Like with Vegeta, his Zenkais are never seen afterwards, no matter how injured he became and healed.
It should be noted that the term Zenkai was neither used in the manga nor the anime. In English, it translates to "complete recovery". Its use as the name for the Saiyan trait is fan made.
Dragon Ball Z
During the Namek Saga and Frieza Saga, Zenkai is an important element of the series, which allows Son Goku and Vegeta's powers to increase in order to "catch up" to Freeza and his men. However, Zenkai are abandoned after the Frieza Saga, but return for a single instance in the Cell arc when Cell returns from his botched self-destruction on the 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.
Outside of the series, Zenkai was also mentioned in Dragon Ball Z: Broly: The Legendary Super Saiyan (it was stated that Broly survived his execution and grew stronger due to his Saiyan genes), as well as Dragon Ball Z: Broly: Second Coming (it was implied that Broly grew significantly stronger while in a coma for seven years after being beaten by Goku and recovering).