Not all FPS games feature "hostage situations" when playing against the computer. In many FPS games, if there is a campaign (player vs. computer), the player follows a storyline and this is what many people play games for or it is the part of the game you hear the most about. This is mainly because it is the one thing that varies from game to game. Multiplayers tend not to have a storyline but is instead a player vs. player with some games allowing only "deathmatch" to be played while others have a variety of game types.
General Multiplayer Game Types:Deathmatch - Where every player fights for himself and attempts to kill all the others. The numbers of respawns (number of times the player can return after being killed) allowed can be adjusted or even removed.
Team Deathmatch - Where there are two teams with the players attempting to kill the people on the opposite team. Each team normally shares the number of respawns.
Capture the Flag - Once again there are two teams. Each team has a flag to defend while attempting to capture the enemy flag. Quite like the game we play in real life, the only difference being that you can kill the other team's members while defending your flag or capturing theirs.
King of the Hill (has a few different names actually) - The map where people play contains a specific object or location. While a player is holding the object or is alone within the predetermined location their personal score goes up. Every player is for themselves.
Deathmatch and Team Deathmatch are normally included in every FPS game (depending on the game). There are many games that only have the campaign mode where the player can fight their way through a storyline. At the same time, there are FPS games which requite no actual shooting. Game such as Portal are actually puzzle games, I'd encourage a search of this game (produced by Valve company) as I've played it myself and quite enjoyed it, but that's getting a bit off topic.
As I said, the storyline is what draws people the most. This is partially why you find many games that have sequels (Halo, Call of Duty, Half Life to name but three of many). The Halo series (one I've recently been reading into) has a long, developed backstory and storyline. I think it has a rather interesting world (our world... but years in the future) and have been enjoying gleaning from it. For many players of the Halo and Half Life games (I can't speak for Call of Duty as I don't really play it) the reason they keep coming back is the story.
Halo features the same MC repeatedly that the player gets to play as (with two of the games being an exception to this). They get to play the hero of earth (and other human controlled planets) many times over fighting off an aggressive and overpowering alien force bent on the destruction and slavery of the human race. There are a multitude of plot twists and amazing discoveries, plus the added fact that the player is always on edge attempting to survive the computer controlled enemy is enticing.
As see, at this point I come to yet another kind of FPS, the Role Playing Game (RPG) kind. RPG games, as opposed to other FPS games that prefabricate your character, allow you to create (within the confines of the game) your own character to play out the storyline. RPG games allow the player a much wider storyline than FPS games. Games like Halo have one storyline that you are always going to follow any time you play the game. RPG games, such as Skyrim, allow for the player to choose which storyline to follow. There are shorter stories, medium stories and the main story to occupy the player in a large playing field. Multiple stories can be followed at the same time and you (partially) get to choose how things go in the end.
Now, I'm going to stop myself before I feel much more like I'm writing an "Airi Post" but feel free to ask me any questions you've got about gaming. I can't say that I play a huge amount of games, but I know my way around

Ps. Communication while gaming:
For online multiplayers there can be a fair amount of communication, but is depends on who you're playing with and if they know the keycode to make audible comments (more FPS games happen too fast for a lot of chatting and voice chat can happen, but not with the games I play). When you play just with your friends (a LAN (local area network] game) more of your communication can happen just by talking with the people around you (since you normally stay in the same room). It tends to more exciting if you play or can vocally chat with the people you're playing with. The human interaction can increase teamwork because people are organized and not just running off to do whatever they think needs to be done.