The Superbat is a term used to refer to Batman during a time when Bruce Wayne had received Superman's Kryptonian powers via a magical talisman known as the Brooch of Cawdor; which is said to grant the greatest desire of the user but at a price.
Gaining all of the typical kryptonian abilities; super strength, invulnerability, flight, super hearing, heat vision, freeze breath, super speed, x-ray vision, among others in addition to his own skills, Bruce was able to patrol and stop all crime in Gotham, causing the first night in the city's history that the GCPD did not receive a 911 call due to Batman being able to detect and prevent all potential crimes through the use of his new powers.
His powers even made it viable for him to track crime entering Gotham back to their source; tracking a new varient of Venom to Bane in Santa Prisca. However, Bruce's new powers gave him a sense of freedom, no longer needing to eat or sleep; causing him to no longer hold himself back due to human frailties. After nearly killing Bane, Bruce realised he could patrol the entire planet and stop crime anywhere.
Warning the Justice League not to stop him, Batman takes on the responsibility of chasing the planet's nightside, fighting crime and protecting the innocents of the entire planet with his new powers.
Concerned with Bruce's actions; that he was no longer restraining himself and that his attitude was quickly developing close to a god complex, Nightwing attempts to stop him but is beaten down. However, during an attempt by the JLA to stop him, Zatanna and a depowered Clark Kent manage to lure Bruce to them and switch his kryptonian powers back to Clark through using the brooch.