A hostage crisis develops when one or more terrorists or criminals hold people against their will and try to hold off the authorities by force, threatening to kill the hostages if provoked or attacked.
Typically, the party of the hostage-taker(s) will issue demands to the forces keeping him/her, or them, surrounded. In a planned hostage crisis, there is often a list of political or religious demands, often including the release of imprisoned friends or allies. In cases where the hostage situation was improvised as a desperate attempt to avoid capture, the demands usually revolve around exchanging the lives of the hostages for transport to safety.
Journalists sometimes use the alternative term siege to describe these incidents. However, events like the Waco Siege are not necessarily hostage crises, because third parties are not being held or threatened.