(covers information from several alternate timelines)
A status report (also known as a condition report) was a type of report which provided an update regarding the current condition of an entity, usually a starship or space station, during or after an event. Technological systems, including weapons and shields, propulsion, and sensors were routinely checked, as well as the entity's crew for casualties.
A status report on an entire ship could also be requested. The computer could give this report and included data on at least the warp core, power systems and levels, structural integrity, shields, weapons, communications, and waste systems. The Doctor, as an unranked Starfleet officer, had access to this report. (VOY: "Projections")
A ship's status report was also one of the responsibilities of the first officer. Chakotay once gave Captain Kathryn Janeway a ships status report after an incident with the Borg, which discussed the operation of removing Borg technology from the USS Voyager. (VOY: "The Gift")
Starfleet away team members were required to file status reports on a regular basis if they were on an away mission for an extended period of time. Naomi Wildman asked Neelix why her mother had not filed a report recently. (VOY: "Once Upon a Time")
The phrase "report" was commonly used by Starfleet captains and commanders as an abbreviation for a status report request, typically when it was unclear what had actually happened. In a situation aboard the bridge, the bridge officers would each state the condition of the systems they are responsible for, and occasionally engineering, sickbay, astrometrics, and other departments would also give a report.
A status report on a specific system, action, or situation was typically requested with the phrase "status."
In 2151, Ensign Travis Mayweather searched the historical archive for information about the SS Conestoga and the colonists founding the Terra Nova colony. He found tons of data including crew manifests, survey photos, and weekly status reports but no sign why the colonists disappeared. (ENT: "Terra Nova")
Examples of status report requests
In 2155 in the mirror universe, Lieutenant Hoshi Sato was the captain's woman of Captain Maximilian Forrest. When she found him in his quarters reviewing status reports, she stated that his quarters were his sanctuary, turned off his monitor, and kissed him. (ENT: "In a Mirror, Darkly")
Admiral James T. Kirk asked Captain Spock for a condition report soon after the HMS Bounty exited a time warp upon returning to the 23rd century in the year 2286, but Spock was unable to provide one since the computer was offline. (Star Trek IV: The Voyage Home)
When Captain Picard asked for a status report of the aligned dilithium crystals, Leland T. Lynch asked for three more minutes. (TNG: "Skin of Evil") Another status report by Worf said that all systems were operational. (TNG: "Symbiosis")
Captain Picard also requested a status report from the computer when the crew's memories were temporarily erased by a Satarran imposter, but the system's voice interface was nonfunctional. Lieutenant commander Geordi La Forge was instead able to pull up a general status report from a panel, which showed that the ship's life support system was fully operational, but navigation, propulsion and outside communications had been completely disrupted. As internal communications were functional, however, Lieutenant Worf asked all ship's personnel to select a representative from their respective groups to report on their status. (TNG: "Conundrum")
After Dukat and Kira Nerys captured a Klingon Bird-of-Prey, they found its computer logs included valuable status reports from several Klingon starships in the Cardassian Union. (DS9: "Return to Grace")
In 2376, the Delta Flyer encountered a dark matter lifeform while Captain Janeway was the lone person in the cockpit section and asked the crewmembers in the aft section for a report on their status. (VOY: "Good Shepherd")