Bioshock is the story of Jack a not-quite-nameless and not-quite-voiceless protagonist, due to the fact he has a name and says a line of dialogue. Why is he "Not-quite-nameless" and "Not-quite-voiceless" then? Surely he must just be Jack, who talks?
Well. No. He is nameless and voiceless in general, as he says a line of dialogue in the opening of the game, and this is the only time he speaks. And this is also the only time that his name is ever used. So for all intents and purposes he is nameless and voiceless. 
Anyway, Jack is flying on a plane to some unknown destination, when he crash lands into the ocean. Fun times. He swims to a nearby lighthouse, which by luck turns out to have a bathysphere in it, which takes him down into Rapture, and underwater haven for intellects and simply hard workers created by a one "Andrew Ryan." After watching a short video explaining what Rapture is, and seeing its beauty and splendor on the way underwater, it comes as a strange shock when the first thing Jack sees upon entering Rapture itself, is some kind of crazed mutant trying to kill him. After this he meets Atlas, who asks him if he would kindly help him out a little in his attempts to escape Rapture with his family.
Basicly. Rapture is a hell-hole. Once, we are told, it was a great city of promise. A haven from both Capitalism and Communism. Many great discoveries were made there, things like Plasmids, which are basicly magic powers gained through the use of drugs, for example. But then everything begins to fall apart. A smuggler named Fontaine, who wants to sell plasmids for profit, ends up in all out war against Ryan, and soon all of Rapture is pulled into it. This all happens before Jack even arrives though, by the point the game starts Fontaine is dead but most of Rapture's citizens are plasmid-powered crazies called "Splicers" who want nothing more than to kill you. That, and the city is leaking. Badly. The fact that the plane you were on happened to crash into part of Rapture sure didn't help either.
The story is told in a variety of ways which all work really well together. You have Atlas constantly nagging in your ear for you to do stuff and informing you about what is generally going on, but you also find old cassette tapes recorded by previous inhabitants of Rapture, adding a lot of backstory to the crumbling city.
The plot over all is very good, it really draws you in and the twist near the end is simply fantastic. It's just a real shame that there is so much filler after that.
Graphically, Bioshock is impressive. It shows that the Unreal Engine 3 can do much more than just overly-macho guys and gals. In fact, most of the enemies in the game are spindly splicers, no macho here. Except for the Big Daddys of course. Big Daddys are large...things...in old fashioned Scuba-gear usually wielding a drill and another weapon of some sort. More on them later though.
The graphics rely heavily on a strange kind of art-nouveau/art-deco/steampunk art style which makes the enviroment look absolutely fantastic. Technically, it isn't that impressive, you'll see blurry textures and a few glitches here and there. And overall the fidelity isn't that great. In fact, the animation in the enemies is some times just down right terrible. But in general the game makes up for its technical short comings through this strange but magnificent style which makes Rapture look fantastic, and vaguely eery. There is always a kind of...anxiety created by the atmosphere of this game, you are always on-edge wondering where the next enemy is going to come from (because Rapture's citizens really like to jump out from hiding spots) and some of the most bizarre residents of Rapture are genuinly pretty disturbing.
Unfortunately, the enemy models all look fairly similar to each other, with a few differences here and there, and don't live up to the standards made by the fantastic enviromental design.
Rapture itself is a fairly claustraphobic place, there isn't much exploring to be done here. Which...sort of makes sense in the context, I mean, you are underwater after all. But it feels like the developers could have opened the world up a little bit by simply adding more rooms that you didn't need to go through on your way through the plot that contained...stuff. I'm not normally the kind of person who is for adding "Open Worldness" into games, Ubisoft has already proven several times over that adding an open world to a game does not instantly make it better, and in fact normally makes it worse, but Bioshock feels like it would have been better if there was at least a little more exploration to do. There is some to do, and you will be rewarded with weapon upgrades and hidden tonics and plasmids for your trouble, but these hidden things are sparce and overall the game feels extremely restricted. You can backtrack, but there is no incentive to do so, you are almost constantly pushed forward through the world.
There is a vague attempt at trying to make you backtrack, through the Big Daddy and Little Sister 'system.' Basicly, you can fight Big Daddys throughout the world and kill them (no easy task, they can take a LOT of hits) and once you do, you free the Little Sister that they were protecting from their custody, and are given the choice to either kill her and take her "Adam" (a resource which you can use to buy plasmids and upgrades with) or to cure her and set her free, getting a lot less Adam. The first choice is the better in the short term, allowing you to gain more Adam quicker and thus upgrade your plasmids and tonics quicker. However, freeing them results in them leaving presents for you later in the game which contain some useful things. This is also the closest thing to an RPG element that the game contains, which if the publishers are to be believed makes the whole game and RPG somehow.
And now we reach the part of the review which talks about the thing most crucial to a game. And also the thing that Bioshock completely fails at. The Gameplay.
Bioshock is a first person shooter, ignore anything on the box which says RPG (I honestly don't know why it says that on the box o.o) because this is not an RPG in any sense of the word. It is an FPS, and for an FPS to be good it needs to have guns which feel like they are actually guns. Which shoot bullets. And thus damage enemies in a way that a real bullet would, and makes the guns feel at least slightly powerful. Bioshock does not have this.
The feel of the guns in Bioshock are terrible. There is no possible way someone could ever go and play something like Crysis or Fear, then play this game and think "Wow, the guns sure are really fun in this!" Basicly, everything feels like a pea-shooter or a BB gun when fighting splicers. Even fully upgraded weapons do hardly any damage to opponents. You can literally headshot a Splicer in the later parts of the game with a Shotgun around 4-5 times before he'll even think about dying. The idea of enemies getting progressively harder to kill in a game is usually balanced with gun upgrades that matter, but not so here. Your upgrades make very little difference to your weaponry, and the ammo for the weapons that aren't completely useless is so sparce and expensive it becomes a chore to defeat enemies. It also doesn't help that enemies barely react to your shots at all. You can shoot enemies wherever you want on their body and it'll make no difference to their reaction. Shooting their head naturally makes them take a little more damage, but they don't even move when you shoot them. Something which makes it feel like you aren't even shooting them at all. Shoot an opponent in the leg and he'll keep running at you (good luck trying to hit them on the leg mind you, the splicers jump around like demented rabbits), shoot him in his arm, he won't even flinch. It really detracts from the experience and makes the combat dull and repetitive.The plasmids try to even this out, as they become fairly powerful towards the endgame, but Plasmids all run off the same supply of power, and if relied on too heavily, you will run out quickly. The game was probably designed this way to give players a reason to balance their use of both plasmids and weapons but it really doesn't work. You'll get generic things like being able to set things on fire, freeze things, shock things, etc. but then you get more interesting things like the plasmid which turns Big Daddys into your ally or one which makes any security cameras focus on whomever you hit with a ball of...something. The plasmids are a lot of fun and it feels like a lot more effort was made into making the enemies react to them than the guns. Zap someone with electricity and they'll spasm in pain and by unable to move, ice will freeze them, fire will cause them to run around trying to pat out the flames and find some water, etc.
Bioshock has a plentiful supply of guns, almost all of which are fairly standard, but their models are pretty unique, and some of the upgrades for them are laughably crazy, but awesome. Such as the upgrade for the standard revolver which adds another clip to it. It LITERALLY just sticks another clip of ammo onto the side of the gun. You get your standard revolver, shotgun, tommy gun, rocket launcher, etc etc. There is nothing too impressive until you get the crossbow. The neat thing about the weapons though is that you can switch out your ammo types to more unique ammo. For most of the weapons this boils down to Regular Bullets, Bullets That Are Good Against People and Bullets That Are Good Against Armour (basicly Big Daddys.) But the rocket launcher can switch between three ammo types, grenades, rockets, and proximity mines. And the flamethrower can switch between flames, electricity and ice, allowing you to burn, stun or freeze enemies depending on your mood.
You can also hack turrets and security cameras and things, which is really useful early on in the game, but practically useless towards the end. But hacking has a minigame that basically just Pipe Mania, and who doesn't love Pipe Mania?
Bioshock also doesn't have much in the way or replay value. After you have played through this game once, you have seen everything there is to see. The sheer linearity of it all means that there is no reason to play it again. Sure, you might want to see the other endings (there are three, based on how many Little Sisters you save/kill) but they are all bad, so probably not. Sure, you might want to explore Rapture more but...there is nothing to explore. This isn't THAT big an issue considering the length of the story but it would have been nice for Rapture to be a bit...bigger. The game teases this giant underwater utopia multiple times, so it would be great to be able to explore it a little further. I'm not saying it should be open-world, but i am saying that the exploration should not be limited to one or two dead-end corridors that contain some loot.
Now, about the genre of the game. The box calls this game an RPG but for the most part I would disagree with it. There is one sole thing that I would consider a good argument for it being labeled as an RPG, and that is the Tonic system. Having to choose how to spec your character by picking different stat altering tonics is very RPGish and the closest Bioshock ever gets to the genre. However a lot of them really feel like they don't make much difference at all if you are good enough at the game, making the choice of tonics a lot less vital.Overall, Bioshock is a great...experience. But as a game it just fails to deliver. The gunplay is mediocre at the best of times and terrible at the worst, the plasmids are fun but limited. The only reason there is for anyone to play this game is for its fantastic narrative and just general atmosphere.