A hostage is a person or an entity that is held by another party. Originally a hostage was a person or entity who was handed over or seized between two parties as a means of providing security for the carrying out of an agreement or as security against one party performing certain acts against another. In modern times, a hostage is often a person seized by a criminal or criminal organization to compel another party to perform certain actions or refrain from performing actions, often under threat of violence to the person being held after a certain amount of time has passed.
In 2268, the Elasi leader Cereth ajElasi and his followers hijacked the Federation vessel USS Masada and took the crew hostage. Cereth attempted to use the hostages he had taken to force the Federation to free a number of Elasi prisoners held by the Federation when the USS Enterprise arrived to investigate the situation. However a landing party from the Enterprise was able to beam on board the Masada, free the hostages, and capture the Elasi hijackers. (TOS video game: Star Trek: 25th Anniversary)
In the early 24th century, Karnas of Mordan IV seized the passengers of a nearby starliner, in order to force the Federation to give him weapons to defeat a rival clan which had assassinated his father. After two negotiators had been killed, then Commander Mark Jameson went to Mordan IV, and was able to secure the release of the hostages. What Jameson had not told anyone was that he had provided the weapons Karnas had demanded - after which Karnas released the hostages. At the same time, Jameson had provided the same weapons to the rival clan - feeling that to allow them to settle their differences on an equal footing was the best application of the Prime Directive in this instance. However, this resulted in a civil war which lasted for over four decades, which killed millions and devastated most of the planet. In revenge, Karnas took the staff of the Federation embassy on Mordan IV hostage in 2364. Jameson - by now an elderly admiral - took an overdose of a de-aging drug to make himself young enough to face Karnas. When Karnas saw Jameson die from the drug, he released the hostages and permitted Jameson to be buried on Mordan IV. (TNG episode: "Too Short a Season")
When the Klingon Empire informed Captain Jean-Luc Picard that the Korris and Konmel - who the Enterprise had recently rescued from the Talarian freighter Batris - were renegades, Picard ordered their arrest and confinement until a Klingon ship could arrive to take them into custody. When Korris and Konmel were confronted by USS Enterprise-D security, Korris grabbed a young Human girl who happened to run out of a nearby turbolift. When Lieutenant Tasha Yar informed the bridge that they now had a hostage situation, Korris quickly released the child to Worf, who in turn handed the child her back over to her mother. After the Klingons were taken away, Lieutenant Tasha Yar expressed her fears that Korris would've held that little girl as a hostage. Worf informed her that was not his people's way, that "cowards take hostages. Klingons do not." (TNG episode: "Heart of Glory")
In 2366, the Rutian Ansata terrorist group took Doctor Beverly Crusher hostage in order to have her treat members of the Ansata who had become ill after using an inter-dimensional transporter. The group later took Captain Picard hostage in order to force a Federation withdrawal from the Rutian system, as well as to force a Federation mediator to come to the system and help the Ansata and Rutian government settle their differences. When their leader Kyril Flynn boarded the Enterprise-D, the crew was able to trace the inter-dimensional transporter signal back to the Ansata base. In a joint Starfleet-Rutian operation, Flynn was killed, the Federation hostages were rescued, and most of the Ansata leadership was captured. (TNG episode: "The High Ground")
In Jarada society, the role of hostages is closer to the original definition. Ritualized hostage exchanges are still performed between a hive and another party - such as another hive. In Jarada society, the returning of hostages is sometimes used as payment for services rendered by another party. The leaders of the Jarada hive Zel took several crew members of the Enterprise-D hostage to guarantee that the Enterprise would help them cure an illness afflicting their population - which was a chemical imbalance caused by a lack of certain minerals in their diets. Once Captain Picard came to understand the role of ritualized hostage exchanges in Jarada society, he cooperated in giving the Zel hive the information they needed to mine other planets in their system as well as providing them with mineral supplements. (TNG novel: Imbalance)