A damage report was a type of report summarizing the damage sustained by a starship or space station to its technological systems, usually due to combat. The commanding officer typically would ask for a verbal damage report immediately after damage is sustained, and also had a written one prepared by the first officer on a PADD.
That same year, when the USS Enterprise-D was caught in a temporal causality loop, Commander William T. Riker requested a damage report after the ship was clipped by the USS Bozeman. This request ended up being repeated with each successive iteration of the loop until the two ships finally broke out of it. Doctor Beverly Crusher reported that there were casualty reports coming in from all over the ship and Lieutenant commander Data reported that starboard nacelle had sustained a direct impact and was venting drive plasma. (TNG: "Cause and Effect")
Picard asked Worf for an immediate damage report after Major "Rakal" ordered the IRW Khazara to fire upon the USS Enterprise-D. Worf reported there was minimal damage as the disruptor beam had little power. (TNG: "Face of the Enemy")
After the USS Saratoga was severely damaged by a Borg cube at the Battle of Wolf 359, Benjamin Sisko ordered a damage report. The tactical officer reported damage to decks 1 through 4. (DS9: "Emissary")
In 2369, when Vash and Q were present on Deep Space 9, a hull breach occurred. Benjamin Sisko asked for a damage report. Kira Nerys reported of a hull breach in an upper bulkhead while Miles O'Brien reported hull fractures throughout the station. (DS9: "Q-Less")
A "full damage report" includes information that is usually part of a status report, including antimatter supplies, and casualties. In 2372, Captain Kathryn Janeway asked for a full damage report, which Tuvok provided in such detail. (VOY: "Deadlock")
In 2373, when their class 2 shuttle crash landed on a Nezu colony planet, Lieutenant Tuvok asked Neelix to inspect its toroidal antennae and provide a damage report. Neelix, eager to help, suggested that he might be able to repair them, but Tuvok insisted that a damage report would be sufficient. (VOY: "Rise")
In 2374, after successfully destroying a Jem'Hadar attack ship, Commander Karen Farris gave Captain Tim Watters a written preliminary damage report. (DS9: "Valiant") That same year, Commander Chakotay requested full damage reports from all decks of the USS Voyager after a translocator was used to steal vital technology from the ship. (VOY: "Concerning Flight")