Bioshock Infinite's ending has been explained, and to do so leads to spoilers. If you don't want the game spoiled, do not read on.
Booker DeWitt may not realize it from the start, but he is actually part of BioShock Infinite's mind-bending plot twist, and Elizabeth is part of a decision he made.
BioShock Infinite is based on the theory of quantum mechanics. To truly understand quantum mechanics may take a bit of heavy thinking, but here's a simple way of putting it: Every decision you make creates an alternate reality in which the opposite decision was made, so in theory, there are potentially millions of alternate realities based on little more than your own choices.
BioShock Infinite is based on what Rosalind Retuce and her "brother" Robert call a thought experiment. They bring Booker DeWitt to the lighthouse and thus the experiment begins. Booker DeWitt's major decision was the baptism at the beginning, which he refused at first. He was supposed to accept it, and the game began its dimension-hopping consequences from there. The fact of the matter is that Rosalind and Robert are the same person from alternate realities.
Rosalind and Robert are actually walking plot devices, and as the game comes to a conclusion, you will notice that they are there to prove that you're stuck in an alternate reality and chance is not an option. A key moment in the game is the coin toss. Booker DeWitt cannot flip the coin and make it land on "tails."
In the end, you have a moment where Elizabeth confronts you from all of her alternate realities, as she is the only one, due to a pinky-severing incident, who can travel between realities. The goal is to eliminate Comstock by agreeing to the baptism. Comstock is the guy who kidnaps Booker DeWitt's daughter Anna, and might actually be you in the future. As a result of the baptism, every instance of Elizabeth vanishes.
At the very end, we see what appears to be Elizabeth wandering the halls of Rapture, giving the ultimate impression that the first game was actually just a result of the ending of BioShock Infinite.
What do you think the ending of BioShock Infinite means?