Elizabeth has been imprisoned in Columbia, a floating city in the sky which exists in the early 1900's, and has been there for 18 years - ever since Elizabeth's childhood. In Bioshock Infinite, Booker DeWitt travels through Columbia to find Elizabeth and help rescue her. Elizabeth possesses the ability to open tears, windows into alternate dimensions, which she uses often. Elizabeth is voiced by Courtnee Draper.
Elizabeth's changes over the course of development
Elizabeth's first appearance was in the Bioshock Infinite trailer where she used her powers to try and save a man falling down through the skies of Columbia (possibly DeWitt), however after reaching her hand out to try and catch the protagonist of the trailer she is dragged off by a large armored arm and her power ceases to function, causing the protagonist to fall.
The first appearance of Elizabeth in actual gameplay was in the ten minutes of promotional Bioshock Infinite gameplay footage released in September 2010. She first appears helping DeWitt defeat a group of angry Columbian citizens before travelling onwards with him and helping him in his fight against the large armored man who seemed to drag her off in the trailer. At the end of the gameplay footage DeWiit asks her if that was the man who imprisoned her but she points to another man standing some distance behind her and DeWitt who then attacks them. Between the original trailer and the gameplay footage a number of changes can be seen in Elizabeth's character design. Major changes were made to her facial structure, her hair color is changed and she appears wearing a white corset as opposed to her original yellow.
Elizabeth appeared again in a trailer for Bioshock Infinite shown during E3 2011. In this trailer she appeared with a softer and rounder face, a more reddened skin, medium brown hair (as opposed to her previous black hair), and a deeper blue dress. The protagonist in the trailer, presumably DeWitt promises that he will stop the man who took her, before she responds that he cannot keep that promise, but that he must keep her from falling into his hands again. In interviews with creative director Ken Levine at the expo, he talked about tears in reality that appeared in the game showing what could have happened should people in the world have taken different choices. He explained that Elizabeth has the power to manipulate these tears but that it is not entirely under her control.