It is the turn of the century. A man named Booker DeWitt is rowed to a lighthouse. He has been hired to go to a city in the sky, Columbia, in order to retrieve a girl. He is a war hero guilty of some pretty brutal stuff in one of the wars between the United States and Native Americans. He then worked for the Pinkertons, a sort of private security force that suppressed unions, among other things. DeWitt enters Columbia. Columbia is a floating city that was built as part of a World's Fair exhibition by a prophet / leader named Zachary Comstock. Comstock used Columbia's arms to attack the Chinese during the Boxer Rebellion. Booker enters the city, which appears to be idyllic but is actually facing civil unrest at the hands of a band of anarchists called the Vox Populi (voice of the people). He rescues the girl he was sent to rescue, the daughter of Zachary Comstock, who is named Elizabeth. She is cooped up in a tower. Elizabeth has the power to open tears in the fabric of reality, presumably to different times or places. She is guarded... or protected? by a giant mechanical bird named Songbird.
Throughout the game, there are things that are obvious anachronistic to the time where the game is set--songs that were composed after the game was written, technology which did not exist at the time, etc. Booker and Elizabeth have a tumultuous connection, and she deplores his use of violence. He escapes with Elizabeth in an airship, following the directions of his clients to deliver Elizabeth to New York City. She attempts to escape, seeing Booker as more captive than liberator.
Columbia's civil war starts in earnest. The leader of the Vox Populi, Daisy Fitzroy, agrees to help Booker if he runs a fetch quest for them. The details of this are not particularly important, but during the course of doing so, Booker and Elizabeth enter into one of her tears in reality and emerge in what appears to be a slightly parallel dimension where the same civil war is occurring, but Booker in this reality is dead. Daisy in this reality has been using Booker as a martyr, and thus wants to stop the living Booker. Elizabeth ends up killing Daisy to save an innocent child. Having grown and lost her innocence, she cuts her hair. Booker and Elizabeth escape on the airship again, only to be attacked by Songbird and crash into Columbia.
Throughout the game, it is noted that two scientists named the Lutece Twins were responsible for much of the invention that had occurred in Columbia. In fact, they are not twins. They are the same person in difference realities; suggesting a sort of quantum divergence where in one reality the child was born male and in another reality female. They each, in their own realities, discovered a way to open a gateway between realities, and crossed into the present reality where they met. While experimenting with the Lutece's device, Comstock is rendered sterile. Despite his sterility, he has a religious vision of the future where his daughter, running Columbia, rains hellfire on New York City. As a result, he kidnapped a child from another dimension--Elizabeth. The source of Elizabeth's powers are that while being brought across dimensions, the portal between dimensions closed, amputating her finger. This made her stuck in between dimensions, and gave her the power to manipulate reality. Comstock imprisoned Elizabeth for this reason and built a device to suppress her powers.
In an attempt to destroy this device and escape, Elizabeth is captured. Booker is drawn into the 1980s where he meets an older Elizabeth who, fulfilling Comstock's prophecy, is destroying New York City. She gives Booker information necessary to shut down Songbird and rescue Elizabeth. He rescues her and chases Comstock. When they meet, Comstock explains that Booker is responsible for Elizabeth's suffering, and allows himself to be killed by Booker. Booker disables the device keeping Elizabeth's powers in check. Songbird attacks the pair. Elizabeth, now at her full powers, draws all three into another dimension.
The other dimension a city that is underwater. Songbird is outside in the water, and drowns. Players recognize this place as Rapture. Booker and Elizabeth take the submarine to the surface and exit the lighthouse that Jack entered in BioShock. But this time, there are hundreds of lighthouses. Connected. There are an infinite number of dimensions, a true multi-verse. And Elizabeth sees all possibilities, and all realities. There is always a man and always a city--in some dimensions, Rapture is built by Andrew Ryan. In others, Columbia is built by Comstock. Columbia's technology was developed in part because a scientist named Fink used the tears in reality to communicate and steal ideas from the technology of Rapture in other dimensions. We also learn that the songs that are played in the wrong generation in Columbia come from other dimensions. Everything is connected. Elizabeth walks Booker through a nexus connecting all dimensions, and explains what's going on, as she now knows what she must do. Booker was Elizabeth's father. She was known as Anna DeWitt (AD on his hand). When she was a girl, Booker racked up gambling debts and sold her to pay them off. When he regretted his choice and tried to stop the Lutece Twins from taking Anna, the ensuing struggle caused the amputation of her pinky as noted, above and set the whole thing in motion. In fact Booker was not hired to save Anna; the Lutece Twins chose to have him do so because they hated what Comstock did to Elizabeth. His memories of being hired were partly implanted and partly made up by his brain to make sense of the parallel dimension he found himself drawn in. There are an infinite number of dimensions, each with different outcomes.
In all dimensions, Booker's rescue of Elizabeth failed, even having killed Comstock. Elizabeth knows that to stop the cycle and truly save herself everywhere, the pair must stop Comstock from ever being born. Elizabeth takes herself and Booker to a baptism scene on a hill--in fact, Comstock is not "born". Comstock is Booker. In some dimensions, he chooses to be baptized and ends up building Columbia. In others he rejects baptism and stays Booker. Elizabeths from every dimension drown Comstock-Booker in the baptismal font, ending all possible realities where Comstock ever existed. Thinking of dimensions as branching off infinitely with each choice made, Elizabeth prunes the tree of possibilities so that none of the events of BioShock Infinite were ever able to happen. Booker awakes to find Anna in her crib.