Zenkai is the power boost that a Saiyan receives once he/she has experienced near death. In the Dragon Ball franchise, this can best be seen when Vegeta receives a large enough boost to over-power Zarbon, an enemy who had beaten him only a few episodes before.
In a much less literal definition, anyone can experience Zenkai. It's just when someone comes back stronger and better than ever from a bad L taken in life. We all have Zenkai.
Person 1: Damn bro I heardLaura rejected you the other day.
Person 2: It's all good man, my Zenkai is kicking in. I'll just hook up with her best friend.
In Dragonball Z, it's the power boost a Saiyangets whenever he recovers from a mortal wound after losing a fight to an opponent.
Zarbon beat Vegeta to an inch of his life, but let him live. Vegeta, having lived the battle, got a zenkai and made a full recovery inside the healing tank of Frieza's ship. Zarbon was killed by Vegeta during their next encounter.