The hostage is usually a civilian who is captured by a criminal and held against ransom, usually at gun point. In games, hostage situations usually demand precision or stealth. If the hostage is killed, it's game over (in most cases).
Hostage situations in games are generally presented to you in two ways. There is the case where the hostages are being held at gunpoint as protection against you, the player. The only way to save said hostages is by taking out the hostage takers through stealth and precision aiming. In Batman: Arkham City, the cowardly criminals hold their hostages at gun point, thinking they're safe from Batman until a well placed batarang hits them in the face. In games like Modern Warfare 3, the stakes are even higher when the hostage is the president of Russia and that there are multiple hostage takers, all guarding the same hostage. These situations are usually solved by shooting all the terrorists in slow motion breach sequences. Another way to deal with hostage takers is through negotiation. In Deus Ex: Human Revolution, telling the hostage taker that shooting the lady he's currently holding at gun point is a bad idea can save said hostage.
Optional Hostages & Taking Hostages
Not all hostages have to be saved. Again in Deus Ex, saving hostages from a bomb is optional. In Modern Warfare 3, there is an optional hostage that you can shoot if you're in a hurry.
Games such as Kane and Lynch and Splinter Cell let the player take hostages and generally use them as human shields. In Splinter Cell, having a hostage can stop enemies from firing while in Kane and Lynch, if you're hostage is dragging you down, you can shoot him or her in the head.