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.
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")