Season Rebirth first appeared in Diablo III right before Season 5. It allowed players to take any of their non-seasonal characters and put them into a brand new season.
You can rebirth up to three non-seasonal characters in one season. Previous to this, players had to create a new character each season.
Season Rebirth solves the problem of running out of character slots. I used Season Rebirth in Season 7 for the first time.