Thank the heavens for Wikipedia.
BioShock is set during 1960, in Rapture, a fictional underwater dystopian/anti-utopian city. The history of Rapture is learned by the player through audio recordings as he explores the city. Rapture was envisioned by the Randian business magnate Andrew Ryan, who wanted to create a laissez-faire state to escape increasingly oppressive political, economic, and religious authority on land. The city was secretly built in 1946 on a mid-Atlantic seabed, utilizing submarine volcanoes to provide geothermal power. Scientific progress flourished in Rapture leading to rapid developments in engineering and biotechnology thanks in part to the brilliant scientists that Ryan brought to the city. One such advancement was ADAM, stem cells harvested from a previously unknown species of sea slug, which were discovered by Dr. Bridgette Tenenbaum to have the ability to regenerate damaged tissue and rewrite the human genome. In cooperation with businessman and mobster Frank Fontaine, they created the plasmid industry, in which customers could enhance their bodies with superhuman qualities. Tenenbaum found that ADAM could be mass-produced by implanting the slugs in the stomachs of young girls ("Little Sisters"), whom they took from orphanages created by Fontaine.
As time passed, the gap between rich and poor increased. Frank Fontaine established charity organizations to support the underclass (something antithetical to Ryan's philosophy). Ironically, his motives were far from benign; his ultimate goal was to use his charity organizations to manipulate the underclass. He also established a smuggling operation to supply citizens with forbidden items from the surface, such as religious material. These, along with his control of the plasmid industry, made him immensely powerful. He tried to overthrow Ryan, but the revolt was violently crushed and Fontaine was reportedly killed. Ryan seized control of Fontaine's plasmid business. Within a few months, a new figure named Atlas rose as the leader of the disgruntled lower class. On New Year's Eve of 1959, Atlas and his ADAM-infused followers began a new revolt against Ryan that spread throughout Rapture. Ryan in turn began splicing his own forces, and his paranoia had reached such a level he was hanging dozens of people, mostly innocent, in Rapture's main square. In order to solve ADAM shortages, the Little Sisters were mentally conditioned to wander the city and extract ADAM from the dead, recycling it into raw ADAM in their bellies after swallowing it. "Big Daddies", enhanced and mentally sterilized humans in armored diving suits, were created by Dr. Suchong, the scientist behind many plasmids, to protect the Little Sisters in their work.
A drawback of ADAM is that a user must take regular infusions or suffer mental and physical degeneration. As the war disrupted production and supply, every ADAM user in the city eventually went violently insane. By the time the player arrives, only a handful of non-mutated humans survive in barricaded hideouts.
At the start of the game, Jack (the player protagonist) is a passenger on a plane that goes down in the Atlantic Ocean in 1960, after ordered society in Rapture has collapsed. After surfacing, Jack finds himself the only survivor of the crash, and swims to a nearby towering lighthouse on an island, where he finds a bathysphere which he uses to descend into the ocean and enter the city of Rapture. An Irishman, Atlas, via the service radio found in the bathysphere, assists Jack in making his way to safety, while Ryan, believing Jack to be an agent of a surface nation, uses Rapture's automated systems and his pheromone-controlled Splicers against him. Atlas tells Jack that the only way he can survive is to use the abilities granted by plasmids, and that he must kill the Little Sisters to extract their ADAM. Overhearing Atlas' words, Dr. Tenenbaum intercepts Jack, and urges him to save the Little Sisters instead, giving him a plasmid that will displace the embedded sea slugs in each Sister. Atlas says his wife and child have been hiding on a submarine and directs Jack towards it. Just as Jack and Atlas reach the bay where it is located, Ryan has it destroyed; an enraged Atlas asks Jack to kill Ryan.
Eventually, Jack confronts Ryan in his office, where the latter is casually playing golf. Ryan reveals a truth that he has pieced together. Jack was actually born in Rapture a mere two years ago, genetically modified to mature rapidly. He is Ryan's illegitimate son by an affair with Jasmine Jolene, a dancer. Ryan further reveals that, after purchasing Jack's embryo, Frank Fontaine designed him to obey orders that are preceded or followed by the specific phrase "Would you kindly..." Jack was then sent to the surface when the war started to put him beyond Ryan's reach. When the conflict between Fontaine and Ryan reached a stalemate, Jack was sent instructions to board a flight with a package and to use its contents, a revolver, to hijack and crash the plane near the lighthouse; enabling him to return to Rapture as a tool of Fontaine. Because Jack was Ryan's son, he could freely use Rapture's bathysphere network, which had been locked out to everyone except those within Ryan's "genetic ballpark". Finally, Ryan has Jack kill him, wanting to die on his own terms. With Ryan's death, Jack realizes too late that Atlas has also been using the trigger phrase to control him. Atlas reveals himself as Fontaine, who faked his death to throw Ryan off his trail and take control of the city, leaving Jack at the mercy of the reactivated security systems. Dr. Tenenbaum and her Little Sisters help Jack escape through the vent system, where he falls and loses consciousness.
When Jack awakens, Dr. Tenenbaum has already deactivated some of his conditioned responses (such as the trigger phrase itself) and assists him in breaking the remaining ones, among them one that would have eventually stopped his heart. When it becomes clear to Fontaine that he is losing control of Jack, Fontaine points out the peculiar fact that Tenenbaum has survived both World War II as a Holocaust victim and the battle in Rapture, insinuating that she has a secret agenda of her own. With the help of the Little Sisters, Jack is able to track down Fontaine. Fontaine, having been cornered, injects himself with vast amounts of ADAM and becomes an inhuman monster. Jack battles Fontaine, eventually prevailing and allowing the Little Sisters to subdue and extract the ADAM from Fontaine, killing him.
Three endings are possible depending on how the player interacted with the Little Sisters, all narrated by Dr. Tenenbaum. If the player rescued all of the Little Sisters (therefore saving their lives), the ending shows five Little Sisters returning to the surface with Jack and living full lives under his care, including their graduating from college, getting married, and having children; it ends on a heart-warming tone, with an elderly Jack surrounded on his deathbed by all five of the adult Little Sisters.
If the player harvested (and therefore killed) all of the Little Sisters, the game ends with Jack turning on the Sisters after defeating Fontaine, presumably killing them all and taking their ADAM. Tenenbaum narrates what occurred, condemning Jack and his actions, voice thick with anger and contempt. Later in the second ending, a ballistic missile submarine carrying a nuclear missile comes across the wreckage of the plane and is suddenly surrounded by bathyspheres containing Splicers. The Splicers kill all hands aboard the submarine and take control of it. If the player saved some of the Little Sisters, but killed a fair few as well, the ending is visually identical to the second one, though the tone of Tenenbaum's voice is a sad one, as opposed to angry.
