BioShock Infinite had one heck of an ending, and given just how much information was thrown at the player in the final half hour or so, some confusion is certainly warranted.
However, Infinite isn't a game with an ambiguous ending. Much to the contrary, the story becomes very neatly laid out once you start to dig into it. If you're getting a nosebleed trying to piece it together though, here's how it all works.
As a full recap of the game could fill pages, I'll only be going over the specifics of the last portion of the game and assuming that you're familiar with the details of the story leading up to this point. And of course, spoilers will follow.
The biggest issue that seems to be confusing people is the revelation that Booker and Comstock are actually the same person. This isn't as complicated as it sounds. The game deals with the idea that there are an infinite number of universes (which plays into the title), each with minor or major variations on the one you're currently in. Each time a choice is made, new universes are created (or already exist, if you can perceive time like the Lutece twins), representing each possible permutation of that choice.
After the massacre at Wounded Knee, a guilt-ridden Booker looks to find redemption at a baptism. At the last moment, he backs out of it though, and continues to live on as Booker Dewitt. However, that moment represents a choice, and some versions of Booker went through with the Baptism, emerging as a deeply religious man. That man takes the name of Comstock, and embarks on a quest to cleanse the world of sin, eventually building Columbia and becoming the Prophet.
This is where things get a little complicated. Comstock is able to rise to power because of a scientist named Rosalind Lutece, a gifted physicist whose work he funds. Lutece develops a way to "peek" into other universes, including ones that have progressed farther or in radically different ways. Using this, Lutece is able to assemble some truly wondrous technology, including using suspended quantum particles to hold Columbia aloft, while Comstock uses it to form "prophecies," allowing him to cement his status as God's chosen one.
Unfortunately, extended use of the machine has its consequences. An unintended consequence of looking through these "tears" in the fabric of the universe caused a sort of dilution of Comstock's own identity, rapidly aging him and rendering him infertile. This is rather inconvenient for Comstock, as his entire life revolves around the prophecy that "The Seed of the Prophet. Shall sit the throne and drown in flame. the mountains of man." In essence, Comstock needs a kid, his kid to be exact, in order to carry out his legacy.
To accomplish this, Comstock and Robert Lutece (Rosalind's counterpart from yet another universe) travel through a tear to a universe where Booker did not undergo the baptism, and instead went on to have a child. Comstock manages to manipulate Booker into giving up his child, revealed to be an infant Elizabeth, and ferries her away to his own universe to be raised in the Songbird protected tower.
Booker, wracked with even more guilt, spends the next two decades in a drunken stupor. Meanwhile, back in Columbia, Comstock ends up killing his wife, as well as the two Luteces, in order to cover his tracks. Unfortunately for him, the would be assassin attempted to murder the Luteces by sabotaging the tear machine. Instead of killing them, it unintentionally spread their consciousnesses across all of space-time, allowing them to exist in all universes and timelines simultaneously.
This allows them to see what happens down the timeline for Elizabeth. Comstock manages to manipulate her into becoming his prophesized "Lamb," and in the future, she uses her tear opening abilities and the technology of Columbia to start an invasion of the entire world. She doesn't plan to stop there though, as she trains an army of quantum shifted soldiers to invade every other universe, converting them to Comstock's vision of humanity. Robert Lutece doesn't like this very much, and threatens to abandon Rosalind if she doesn't agree to help him put a stop to it.
The two bring Booker into a universe where Comstock and Columbia exists, and begin experimenting with a series of constant and variable situations, trying to guide him into a situation where he rescues Elizabeth. This proves difficult, and most of these Bookers fail or perish along the way. Finally, the Luteces are able to find the correct path that allows Booker to kill Comstock, save Elizabeth, and apparently, escape Columbia.
But wait you might say, how come Booker doesn't remember meeting the Luteces? This is explained in the very first line of the game, "The mind of the subject will desperately struggle to create memories where none exist..."
It's elaborated on throughout the story, but basically it comes down to this: when brought into a new universe, your mind basically cracks under the pressure of trying to reconcile your old memories with the memories that exist in this timeline. Without the proper guidance (as in the case of Robert and Rosalind Lutece), your brain will basically try its best to rationalize the timeline, filling in memories where none exists to form a coherent story. Booker basically invents a scenario where he's rescuing this girl to wipe away a gambling debt, using real-life memories as a foundation; a story that unravels over the course of the game.
Then how about the very last scene of the game, as a number of Elizabeths appear and drown Booker? This bit is a little messy, as it's difficult to tell what is literal and what is representative. Either way though, it boils down to this: with the siphon destroyed, Elizabeth is able to freely exist outside the boundaries of linear time. She realizes that even if one Comstock is killed, a million million more exist in alternate universes, and that the only way to destroy him for good is to "smother him in the cradle." This isn't a literal cradle, it's the idea of going back to Comstock at his birth: the baptism. If Booker is killed before the choice to undergo the baptism is even made, there can exist no timelines where Comstock is born.
Booker realizes this, and allows Elizabeth to drown him, thus ending every timeline where Comstock creates Columbia. In the stinger scene, we see that a universe in which Booker and the infant Elizabeth are still together, giving us hope that while Comstock is dead, Booker and Elizabeth live on.
It might sound a bit dense, but BioShock Infinite is a very deep and richly told game. We are merely scratching the surface here, but hopefully that clears up the ending's specifics a little bit. If you still find yourself scratching your head, perhaps a second playthrough is in order! Or, let me know in the comments section if you'd like to see another article with a few more questions answered!