(covers information from several alternate timelines)
A potential hostage situation
The bridge crew held hostage
A hostage was a person who was held captive as leverage.
In 2151, Enterprise NX-01 captain Jonathan Archer and a landing party visited the Vulcan sanctuary of P'Jem and were accosted by an Andorian faction who believed the facility to be a listening post. Commander Trip Tucker described their situation to Malcolm Reed aboard Enterprise as being a hostage situation. (ENT: "The Andorian Incident")
In August of the same year, Matthew Ryan took a Nausicaan pirate as hostage aboard his Earth Cargo Service freighter ECS Fortunate and held him guarded in a cargo module. (ENT: "Fortunate Son")
Two Romulan officers were held as exchange hostages aboard the USS Enterprise when Captain Kirk and Spock were beamed over to Sub-Commander Tal's flagship. (TOS: "The Enterprise Incident")
The Troglyte Vanna was able to take Captain James T. Kirk hostage for a short time and acquired his Starfleet type 2 phaser. (TOS: "The Cloud Minders")
When the USS Enterprise received no word from Captain Christopher Pike after he was beamed aboard the Narada, acting captain Spock classified him as hostage of the war criminal Nero. (Star Trek)
In 2269, the Shore Leave Planet's computer took Nyota Uhura hostage to force the "sky machine" that was her master to remain. It subsequently decided to "turn off" the landing party that was coming to rescue her, saying it had no need for more hostages. (TAS: "Once Upon a Planet")
Later, Harry Mudd briefly took Christine Chapel as a hostage to escape the Enterprise. (TAS: "Mudd's Passion")
Also that year, Chuft-Captain took Nyota Uhura hostage in exchange for the Slaver weapon. (TAS: "The Slaver Weapon")
When the two Klingons Korris and Konmel met a mother and her daughter in a corridor during a tour led by Worf, security chief Natasha Yar reported a possible hostage situation when one of the Klingons raised the girl. It was equalized when the girl was handed to Worf who gave her back to the mother. Afterwards, when Yar expressed concern about the situation, Worf told her that taking hostages was not their way, that "Cowards take hostages. Klingons do not." (TNG: "Heart of Glory")
The Ornaran T'Jon shortly took Commander William T. Riker as hostage by using his natural electricity. (TNG: "Symbiosis")
In 2366, Geordi La Forge suggested that Kevin and Rishon Uxbridge might be hostages of the hostile force which had destroyed almost all of the Delta Rana IV colony, though Doctor Beverly Crusher indicated that their tricorder readings indicated that they were not under the type of stress that would normally accompany such a situation. (TNG: "The Survivors")
That same year, when Beverly Crusher was captured by Ansata terrorists on Rutia IV, it was suspected that she was wanted as a hostage in order to force Federation involvement in the conflict. She was, in fact, actually captured in the hope that she might be able to treat the eventually fatal effects of the inverter device used by the group. Later, however, the group tried to use both her and Captain Jean-Luc Picard as hostages. (TNG: "The High Ground")
In 2367, in the panic surrounding the supposed return of Ardra to the planet Ventax II, members of a Federation observation team were kidnapped as hostages. When a con artist posing as Ardra learned of this, she ordered their immediate release, stating to Captain Jean-Luc Picard that they need not "cloud this with bit players." Acost Jared, a Ventaxian leader in awe of her, stated that her order would be carried out immediately. (TNG: "Devil's Due")
In 2368, the spirits of Ux-Mal criminals took over the bodies of Counselor Deanna Troi, Chief Miles O'Brien, and Data and took a group of hostages in Ten Forward. (TNG: "Power Play")
In 2369 a group of Klaestrons tried to abduct Jadzia Dax from Deep Space 9 which Odo described as a hostage situation to his security guards. (DS9: "Dax")
In an alternate timeline in 2371, Lieutenant Tom Paris and Captain Kathryn Janeway were abducted by a group protesting the use of polaric ion energy on a planet in the Delta Quadrant. The group's leader, Pe'Nar Makull, instructed Janeway to pose as a government official to help them gain entrance to sabotage a local power plant. Janeway instead advised a guard at the plant that she was a hostage and that the group was attempting to break into the plant. (VOY: "Time and Again")
That same year, Lieutenant Tuvok held the bridge crew hostage when he was possessed by the Komar. (VOY: "Cathexis")
In 2373, a pah-wraith took possession of the body of Keiko O'Brien during her visit to the Fire Caves on Bajor. Upon returning to O'Brien's quarters on Deep Space 9, the pah-wraith informed Keiko's husband Miles that it would hold her body hostage until he had done everything it asked of him. (DS9: "The Assignment")