A drumhead trial was the name for an archaic form of court martial practiced on Earth in the 19th century. Its name derives from the eponymous usage of a drum's head as a seat for the presiding judge in an ad-hoc military tribunal. Such trials were marked by an emphasis on expediency and a notable lack of due process and impartiality. Often, the actual "trial" was a mere formality, with the verdict known well in advance. They were sometimes used to curb similar behavior in other potential offenders. Trials akin to the drumhead-type were conducted in later eras and by different cultures.
The Dramen justice system was swift and conclusions often foregone. This caused Captain James T. Kirk to refer to them as a "kangaroo court" system – "kangaroo court" was a moniker applied on Earth to justice-miscarrying processes similar to drumhead trials. (TAS: "Albatross")
Admiral Satie's "drumhead"
Captain Jean-Luc Picard once cautioned Lieutenant Worf about the ease with which conspiratorial investigations, especially when the alleged conspiracies were sufficient to cause great alarm, could spiral into quasi-drumhead trials. As he explained, the term originated from military tribunals when "military officers would upend a drum on the battlefield. They'd sit at it and dispense summary justice; decisions were quick, punishments severe, appeals denied. Those who came to a drumhead were doomed."
Picard said this in the aftermath of the ad hoc investigations panel, headed by Rear Admiral Norah Satie, who used unfair and frenzied interrogation methods to exceed her mandate and attempt to "bring down" presumed saboteurs and, what she thought, a grand conspiracy on board the USS Enterprise-D in 2367 – which she continued to press even after the lone criminal had already been caught and all other events accounted for. Only the intervention of Admiral Thomas Henry could bring her "drumhead trial," which by then had gone so far as to bring great humiliation to one crewman and accuse Picard of disloyalty, to an end. Picard's talk with Worf was prompted by the latter's remorse for his initial enthusiastic cooperation with Satie's unfair panel. (TNG: "The Drumhead")
Other similar systems
The entire Cardassian "justice system" routinely conducted "trials" akin to drumheads; for them, the accused was always treated as guilty, with no chance of proving innocence, and the "trial" itself was a mere showcase to bring a "lesson" to the masses of crime "not paying," as well as to maintain a sense of the infallibility of the government's legal system. The job of the "prosecutor" was not to prove guilt, and that of the "defense counsel" was not to defend the accused but instead to convince and assist him or her to accept and publicly acknowledge predetermined "guilt." (DS9: "Tribunal")