A Damage report is a type of report summarizing the damage sustained by a starship or space station to it's technological systems, usually due to combat. The commanding officer typically asks for a verbal damage report immediately after damage is sustained, and also has a written one prepared by the first officer on a PADD.
Commander William T. Riker prepared a written damage report for Captain Jean-Luc Picard's review in his ready room after the incident with the Tamarians. (TNG: "Darmok")
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 distruptor beam had little power. (TNG: "Face of the Enemy")
In 2374, after successfully destroying a Jem'Hadar attack ship, Commander Karen Farris gave Captain Tim Watters a written preliminary damage report. (DS9: "Valiant")