Regeneration is one of the possible challenges (also known as "modifiers" and commonly called "handicaps" due to confusion with Challenge Mode) encountered in standard, challenge mode or Survivor battles.
Your opponents will regenerate by 1% (with some variation) of their maximum health every second. Unlike Radiation, all enemy characters will constantly regenerate, even while tagged out. Killer Frost/Regime's passive has no effect on Regeneration, although it used to work.
Batman/Flashpoint is a must, due to his ability to turn off any aspects of healing, disabling even tagged-out opponents from regenerating. Batman/Flashpoint can also counter the enemy regeneration via his "Destructive Blows", which can KO opponents quicker.
Hawkgirl/Prime's snare can prevent enemies from tagging out, limiting other enemies at full health from joining the fight. Her tag cooldown reduction can also allow her to easily switch to a high-damage character while the enemy is snared.
If Cyborg/Teen Titans is on the opposition, it will be a challenging battle, due to his passive stacking with the regeneration "handicap". If this occurs, it is advised to use Batman/Flashpoint.
It is advised to bring Catwoman/Arkham Knight, due to her DoT passive. Because her DoT cannot be rinsed, she can easily overcome regeneration with her DoT.